The definition of "I-stems" is relevant
Although the question said to ignore i-stems, I think it's actually necessary to discuss them, because many Latin nouns of the third declension have a mix of i-stem and consonant-stem forms (to the extent that the division of nouns into these two categories is fairly problematic). And i-stem nouns, as the name implies, can be thought of as having a stem ending in I, which is not a consonant. So to answer this question, it's necessary to resolve the issue of which nouns should be thought of as having stems ending in I.
I-stems and consonant-stems always take the same endings in the dative singular (-ī) and dative/ablative plural (-ibus), so those forms can't be used to distinguish i-stem nouns from consonant-stem nouns.
I-stems and consonant-stems almost always take the same endings in the nominative masculine/feminine plural (-ēs); it is uncommon and unetymological for an i-stem noun to take -īs in the nominative plural.
Accusative singular -em and ablative singular -e forms, athough they seem to have originated as consonant-stem forms, are so commonly found with nouns that are otherwise i-stems that I think that these endings are not considered to be unique to consonant-stem nouns in the Classical Latin time period. But it seems consonant-stem nouns never take -im in the accusative singular.
Because of the somewhat complicated formation of Latin nominative singular forms in the third declension, it can be difficult to analyze the nominative singular form in terms of a root + suffix. We can often figure out the origin of the form from comparative linguistics or internal reconstruction, but the etymological analysis may not always be the one that makes the most sense from a synchronic perspective. A big issue is that a great many nouns that are clearly i-stems in terms of etymology show syncope of the original stem-final vowel in the nominative singular.
A fair amount of third-declension nouns take the i-stem ending -ium in the genitive plural. My understanding is that the presence of -ium in the genitive plural is enough to categorize a noun as being at least partially an i-stem. This raises the issue of whether nouns that show mixed behavior should be analyzed as having multiple stems, with one stem ending in a consonant and another stem ending in -i.
In this post, I will take the most conservative interpretation of your question and only include nouns that have attested -C-um genitive plural forms. (Or when those are exceptionally difficult to find, I may resort to a neuter nominative/accusative plural in -a as an example of a form that is clearly built on a consonant stem.)
Most consonants can occur at the end of a noun stem
It is possible for a noun stem to end in almost any consonant. It's easier to list the consonants that don't appear at the end of noun stems.
The least likely consonants in this position are /j/ (consonantal I), /h/ (H), /z/, and /f/. All of these sounds have particularly restricted distributions in Classical Latin, and it's even a bit dubious to categorize some of them as part of the inventory of normal Latin consonant sounds: /h/ doesn't act like a consonant for purposes of syllable weight or vowel shortening in hiatus, and /z/ only occurs in borrowed words from Greek.
No genitive plural is built on a noun stem ending in /j/
I know of no noun with a genitive plural form ending in -jum.
A gap in nouns with stems ending in /j/ is fairly easy to explain in terms of historical sound changes. In Latin, the consonant /j/ is mostly restricted to two positions: word-initially (also in prefixed or compound words built from a word starting with /j/), or in an intervocalic geminate /j.j/. A sound change is thought to have regularly eliminated original singleton /j/ between vowels (as in trēs from PIE *tréyes).
Traditionally, "ae" and "oe" are analyzed as diphthongs, but even if you analyze them as VC sequences /aj/, oj/, I don't know of any noun with a stem ending in "ae" or "oe". (There are nouns with stems ending in -aea- /ajja/ or -aeo /ajjo/, but those belong to the first and second declensions respectively.)
I don't think there are any likely sources for a borrowed j-stem noun either. Ancient Greek seems to have some nouns with stems that (from an etymological perspective at least) end in -οι- (Wikipedia gives the example ἠχώ), but the /j/ is lost through contraction in all forms except for the vocative singular, so it seems impossible for Greek to be a source of Latin nouns with stems ending in /j/.
There are no H-stem nouns, except maybe vehēs, whose stem could be vehi- instead
In Latin, /h/ is rare outside of morpheme-initial position. It did occur medially as a reflex of PIE g(j)h in some words, but the list seems to be short (the WordReference discussion "PIE *gh > latin [h] word initially and [g] word internally but why veho:, traho: and incoho: with h?" mentions the following words only: mihi, vehō/vehis, cohum, trahō).
H doesn't act like a consonant for the purposes of certain phonological rules: in V̆ChV sequences (between words or produced by combining a prefix ending in V̆C with a base starting with hV) the first syllable scans as short (as in V̆CV sequences), and in VhV sequences the first vowel is regularly short (as in VV sequences) and there is sometimes the possibility of contraction to one syllable.
Latin might have one noun stem ending in /h/, depending on how you view nouns that are at least partially i-stems. The noun vehēs/vehis has an /h/ followed by -ēs or -is. Nouns with these endings are thought to etymologically derive from i-stem nouns, and are traditionally categorized as such, but many nouns in this category can actually take consonant-stem endings like ablative singular -e or even genitive plural -um. I didn't find any examples of either vehium or vehum in the PHI Latin corpus, but vehum is used for the genitive plural in the Codex Theodosianus (mentioned in Roby 1881, p. 141; however, he still categorizes the noun as having a stem ending in -ehi, not in -eh). I haven't found an example of ablative singular vehe in any source.
As far as I know, Greek has no stems ending in /h/. ("Interaspiration" in Greek seems to only be attested in the word ταὧς, and possibly in compounds/prefixed words.) Greek has stems ending in φ, θ, χ, but these are certainly best analyzed as consonants of their own, rather than as clusters ending in /h/. The proper analysis of "ph", "th", "ch" in Classical Latin is a bit less obvious, but for the purposes of this answer I will also treat Latin "ph", "th", "ch" as single consonants (with digraph spellings), rather than as sequences ending in /h/.
No commonly encountered noun stem ends in F. But a third-declension -fēs or -fis noun seems possible in theory
The gap in /f/-stem nouns seems to be a predictable result of historical processes, but not necessarily an active restriction in Latin morphophonology. I think borrowed words would most likely be able to have a stem ending in /f/, although the formation of the nominative singular specifically would raise a phonological problem. The consonant cluster /fs/ is unprecedented in Latin words, and I think it's very unlikely a borrowed word would be given a form ending in the cluster /fs/ in the nominative singular. However, as mentioned above, the "historically i-stem" nominative singular endings -is or -ēs sometimes appear on the nominative singular of masculine or feminine nouns that otherwise inflect with consonant-stem forms, and constructing the nominative singular this way would get around any phonological restriction against word-final or pre-consonantal /f/.
After ph merged with f, examples of borrowed Greek ph stems could possibly be used as examples of a marginal category of Latin /f/-stem nouns; those kinds of borrowed words show -ps in the nominative singular, and <ph> or <f> /f/ before a vowel. (For specific examples, see the section below on "Stems ending in unusual consonants".)
The gap in /f/-stems, like the gap in /j/-stems, is pretty well explained by the history of Latin sound changes. F in native Latin words is almost completely confined to the start of morphemes as the result of certain historical sound changes, so the gap in F at the end of noun stems can also be seen as just a specific case of a more general pattern to the distribution of this consonant sound. The existence of morpheme-medial F in non-native Latin words like rūfus and būfō shows that the paucity of F in non-morpheme-initial position is not enforced by any absolute synchronic constraint that was active in Classical Latin.
No commonly encountered noun stem ends in Z. But a third-declension -zēs or -zis noun seems possible in theory
Z is absent in general from native Latin words (as Joonas mentioned), so the Z gap is not really something specific to stem-final position.
As with /f/, there would be a phonological problem with just adding the suffix /s/ to form the nominative singular of a masculine or feminine noun: /zs/ would be a very bad ending for a Latin word. But as with /f/, this could be avoided by using the nominative singular ending -ēs or -is, as long as you don't view these endings as being exclusively characteristic of i-stem nouns during the time of Classical Latin. There do seem to be examples of names taken from Greek with a nominative singular ending in -zēs and a genitive singular ending in -zis (see the section below on "Stems ending in unusual consonants").
I'm guessing you didn't intend to include invariable nouns or names in your question. The Vulgate does provide examples of invariable borrowed names ending in /z/, such as Achaz 'Ahaz'.
No noun stem ends in a consonant spelled with the letter K, but it doesn't really correspond to a distinct consonant sound of its own
I agree with Joonas that the letter K seems to never be used to spell a stem-final consonant, but I wouldn't consider this to be a true gap, as K was just a variant spelling for the /k/ sound that was usually represented by the letter C.
QU might have one example in torquēs/torquis (but it's a historical i-stem)
If QV is analyzed as a replacement spelling for a CV /kw/ cluster, then it wouldn't count as a consonant of its own, so we wouldn't have to be concerned with whether any stem ends in it. But QV is sometimes analyzed as a consonant of its own (/kʷ/), so I'll cover it as well.
Initially, I agreed with Joonas about the lack of stems ending in Q(V), but then I found the example of torquēs/torquis 'necklace, ring'. The linked L&S entry shows the use of torque as an ablative singular: "“avis torque miniato in cervice distincta,” Plin. 10, 42, 58, § 117".
Like vehēs, torquēs seems to have no genitive plural attested in the PHI corpus. I found one postclassical and non-native example of torquum: the article "Anglo-Norman Historiography and Henry of Huntingdon's Translation of 'The Battle of Brunanburh'", by Kenneth Tiller (2012), cites a translation into Latin from Old English made ca 1123–54 by Henry of Huntingdon as using the wording "nobilibus torquum dator".
Note that the i-stem genitive plural torquium is also attested in post-classical sources.
Examples for the rest of the consonant sounds
The examples for C, D, G, L, M, N, R, T, V in Joonas's answer all work.
P: in the genitive plural, apis can take either the consonant-stem form apum or the i-stem form apium. The form apum, which is clearly built on a stem ending in P, does seem to be more common (Zumpt 1836, p. 28). Another example would be *ops, which forms opum.
B: plēbs takes the i-stem form plēbium. There are certainly many nouns that take -bem in the accusative singular and -be in the ablative singular; I had some trouble finding a third-declension noun with a genitive plural form ending in -bum, but I think Arabs (from Greek Ἄραψ) may count as an example, since it seems to have as its genitive plural the form Arabum.
S: For S, it's hard to find a good example because original s-stem nouns generally turned into r-stems in Classical Latin because of rhotacism. Despite having ossium as the usual genitive plural form, os does clearly show a consonant-stem form in the nominative/accusative plural ossa, so it's not a bad example. The noun as, assis usually takes the i-stem genitive plural form assium, but a consonant-stem form assum is also supposed to be attested.
X: I don't know of any pure consonant-stem third-declension nouns with a stem ending in X. That said, X is just a replacement spelling for CS or GS, so I wouldn't necessarily treat X as a distinct consonant: it's a cluster ending in S.
H: As I mentioned above, I would say that Latin has no H-stem nouns, except for (possibly) vehēs. If we distinguish PH, CH, and TH from P, C and T, I would consider each to be a single consonant of its own that does not contain the consonant phoneme H. PH-, CH- or TH-stem nouns would probably only be possible as loans from Greek: see below for possible examples.
Stems ending in unusual consonants (mostly from Greek):
PH: Gryps, from Greek γρύψ, might count as a PH-stem. I don't think the genitive plural form can be used as a criterion because it's not clear what it is: Zumpt 1836 says that the genitive plural of gryps "does not occur" (p. 29).
CH: sardonyx, from Greek σαρδόνυξ, seems to decline as a CH-stem.
TH: Tiryns, Tirynthis, from Greek Τίρυνς, seems to decline as a TH-stem (Zumpt 1836, p. 21). Since it is a proper noun, I don't think any attested genitive plural form exists.
F: If you include Ecclesiastical Latin, than the word scinifes/cinifes might count, as it apparently has a genitive plural form (s)cinifum. A variant spelling with ph also seems to exist.
Z: If you include proper nouns and don't worry about the issue of i-stems vs. consonant-stems, the name Gōtarzēs (from Greek Γωτάρζης) would probably count. It seems that in Latin, it has a third-declension genitive form Gōtarzis that could be seen as being built on the stem Gōtarz-. (I asked a separate question about the declension of masculine nouns like this from Greek nouns ending in -ης.)
Zumpt, C.G. 1836. Grammar of the Latin Language.