Very few Latin words contain "ei" as a diphthong. Some possible examples are deinde, dein, deinceps, rei, spei, and in fact, the pronoun ei (but not always).
The exact list of examples depends on what you call a "diphthong" (e.g. whether you include words with intervocalic geminate semivowels, or words where the vowels are pronounced in one syllable because of "synizesis").
Old Latin had a diphthong ei (original ai and oi turned into it when they were in non-initial syllables), but because of sound changes, words that had ei in Old Latin ended up being pronounced in Classical Latin with /iː/, the same sound as the reflex of Old Latin long i. (In some cases, ei instead merged with e.)
Because of that sound change, the digraph <ei> after the Old Latin period was used as a spelling of the monophthong /iː/, not only in cases where /iː/ etymologically originated from a diphthong (as in the second declension masculine nominative plural), but also in some cases where /iː/ etymologically originated from a long monophthong.
This historical sound change and the resulting spelling convention where <ei> = /iː/ creates some additional difficulty with determining the pronunciation of Latin word forms spelled with <ei>.
Before a consonant: deinde and friends
Cser (2016) argues that Latin has no genuine diphthongs, only vowel + glide sequences. Cser says that, if we set aside words with geminate /j.j/, /ej/ occurs in the following three related words: deinde, dein, deinceps (p. 32). As far as I know, these are never written with <ej>. L&S says that dehinc is frequently a monosyllable in poetry, which could be interpreted as implying a pronunciation /dejnk/.
Scansion alone doesn't tell us the exact pronunciation of <ei> here. I'm not sure what basis Cser and other scholars have to state that these words had /ej/ specifically, as opposed to some other monosyllabic sound like /eː/, /iː/, or even /e/ or /i/ (with the syllable being heavy because of coda /n/).
Some sources seem to categorize the disyllabic pronunciation of deinde and/or dehinc as an example of synizesis, alongside e.g. the pronunciation of "eo" in one syllable in words like alveo. Synizesis is also supposed to be possible in the word proin (pro͡in = [projn]?), a shortened version of the similarly formed proinde.
Adams (2016) gives dende as a 2nd-century spelling of deinde (Chapter 22, "Letter of Claudius Terentianus (P. MICH. VIII.471, CEL 146), of the Early Second Century"; the text is sourced from Kramer 2007).
In the notes, Adams says that "the phonetics [of "synezesis" in deinde] tend not to be addressed", and that "it remains unclear in what way (or ways) the term was pronounced in the early period; for the scansion dĕīnde in Terence see Questa (2007: 439)."
Adams suggests that a form with ē developed eventually in Latin, bringing up cōgo from co- + ago as an example of a contraction resulting in a long vowel with quality of the first component. As evidence against reading the first syllable of <dende> as [dɛn], Adams brings up (obsolete) reflexes in Spanish and Portuguese with forms like dende: these point towards Proto-Western-Romance *e rather than *ɛ, because *ɛ would lead to the form *diende [djende], which is apparently not attested ("information from Adam Ledgeway"). Adams says the same would go for Spanish dentro (not *dientro) from Latin de- + intro.
Adams doesn't seem to mention that Proto-Western-Romance *e corresponds not only to Latin ē, but also to Latin ĭ, which I think is relevant because de + ĭ > dĭ seems like a plausible alternative kind of contraction.
Word-final [ej] (sometimes?): ei; rei, spei; Pompei
There are a few situations where word-final [ej] may have occurred, but I have the impression that it was often in variation with disyllabic sequences, rather than the only pronunciation used.
The pronunciation of ei, the dative form of is, is a complicated question, but there seem to be cases where it must scan as one syllable (Weiss 2010).
In the cases where it does so scan, it seems likely that the pronunciation was /ej/, which would be favored both by intra-paradigmatic analogy with the genitive singular /ejjus/ and inter-paradigmatic analogy with the datives cui and huic, whose monosyllabic pronunciations are currently widely interpreted as having been /kuj/ and /hujk/, containing falling diphthongs or /j/-offglides.
fifth declension rei, spei
Genitive and dative singular forms of the fifth declension can end in <ei>. The use and pronunciation of these forms is complicated by the existence of a lot of variation in this area; the fifth declension was the smallest Latin noun declension and was subject to various pressures of (attempted) regularization or of analogy with the first declension.
Per Lindsay (1896),
[Because the fifth declension followed the pattern of the first, we should expect] to find find a disyllabic -ēī, which through the working of Latin phonetic laws would become -ĕī, and in rapid utterance even -e͡i (class -ī) in the genitive, but a diphthongal -ei which would become either -ē or -e͡i (class. Lat. -ī) in the dative.
In other words, based on what Lindsay writes, a historical [ej] in these forms should be expected to have developed to a monophthong -ē or -ī.
Exon (1905), who cites Lindsay (1896) along with Seyffert, specifies that in early Latin, the monosyllabic ei at the end of dative fifth declension nouns can undergo elision, while the ei at the end of genitive fifth declension nouns cannot: he argues that this implies that the dative ended in a monophthong intermediate between /iː/ and /eː/ (in Exon's notation, "ẹ̄", which in modern IPA would be /e̝ː/; this is standardly reconstructed as an intermediate step in the shift from Old Latin ei to /iː/~/eː/), whereas the monosyllabic genitive ending ei was an actual phonetic diphthong /ej/ (pages 556-557). Then at a "comparatively late" stage, the disyllabic dative ending came into use.
I don't know how the situation evolved past then, but I think it's possible that some classical Latin speakers ended up using [ej] in either the genitive or the dative based on later compression/synizesis of disyllabic eī or ēī.
genitive Pompei (usually trisyllabic)
It seems that the vocative form Pompei was most commonly pronounced with three heavy syllables (for which see below), but some sources indicate that it also had a pronounciation with two syllables. Allen 1890 cites Horace Carmina II. 7.5:
Pompe͡i meorum prime sodalium.
(This is the first line of an Alcaic stanza, and thus should scan as – – u – – | – u u – u –).
In contrast, Exon 1905 denies knowing any examples of Pompe͡i (page 560).
Priscian writes about this name, but Allen and Exon disagree about what he says.
Before a vowel: /ej.j/
Latin had a number of words where an intervocalic letter <i> represented /j.j/, a geminate palatal approximant/glide/semivowel. It is inferred that these words had a geminate consonant based on the existence of some older spellings with double <ii>, the heavy weight of the syllable preceding the intervocalic <i>, and the development in Romance languages, such as the geminate affricate in Italian maggiore from Latin maiorem. These words have modern spelling variants with <j>.
The geminate palatal approximant /j.j/ could occur after /e/, as in the word eius/ejus mentioned in Joonas's answer.
I would not consider the /ej/ in /ej.jus/ to be a diphthong any more than the /ju/ is.
Some sources, particularly older dictionaries like L&S, may write this as ēi or ēj, where the macron represents syllable weight but does not necessarily represent the true length of the vowel itself.
Some instances of <ei> are thought to have been pronounced /ej.ji/. Cser mentions reicere (p. 149) and the genitive form Pompei (p. 13).
There are some prefixed words where, based on etymology, we would expect /eː.j/, although metrically there's no way of distinguishing this from /ej.j/. You mentioned eicio, which is thought to have started with /eː.ji/; another word that is thought to have had /eː.j/ is seiungo.
(For more discussion of pronunciations like /ji/, /j.ji/, or /j.j/ for <i> in contexts other than just <ei>, see my answer to When is an I not an I?).
other /ej/ before a consonant?
There might be a few other examples of /ej/ before a consonant, but it's not necessarily clear. Cser brings it up as a potential pronunciation of <ei> in "anteis", "anteit" and "anteire" but suggests that /iː/ is a more likely pronunciation in this word (p. 150).
Allen, Frederic D. 1890. "Gajus or Gaïus?". Harvard Studies in Classical Philology, Volumes 1-2.
Lindsay, W. M. December 1896. "The Dative Singular of the Fifth Declension in Latin." The Classical Review, Vol. 10, No. 9, pp. 424-427.
Exon, Charles. 1905. "The Latin Genitive in -ai." Hermathena
Vol. 13, No. 31, pp. 555-574.
Weiss, Michael. 21 September 2010. "The Scansion of dat. sg. ei in Classical Latin", OHCGL Addenda and Corrigenda (blog).
Adams, J. N. 2016. An Anthology of Informal Latin, 200 BC–AD 900: Fifty Texts with Translations and Linguistic Commentary.
Cser, András. 2016. "Aspects of the Phonology and Morphology of Classical Latin."