Very few words contain "ei" as a diphthong.
The exact list of examples depends on what you call a "diphthong" (e.g. whether you include words with synizesis or geminate semivowels).
Possible examples include deinde, dein, deinceps, rei, spei.
The Old Latin diphthong ei (which was the outcome of older ai and oi in non-initial syllables) ended up being pronounced in Classical Latin as /iː/, merging with the reflex of long i (in some cases, it instead merged with /eː/). Because of this, the digraph <ei> was used after the Old Latin period as a spelling for the sound /iː/, both in cases where /iː/ etymologically originated from a diphthong (as in the second declension masculine nominative plural) and in some cases where /iː/ etymologically originated from a long monophthong. This creates some additional difficulty with determining the pronunciation of 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 wouldn't tell us the exact pronunciation of <ei> here, so I'm not sure why Cser thinks these words had /ej/ specifically, as opposed to something 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.
An Anthology of Informal Latin, 200 BC–AD 900: Fifty Texts with Translations and Linguistic Commentary, edited by J. N. 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.
In word-final position: 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.
According to "The Dative Singular of the Fifth Declension in Latin", W. M. Lindsay, The Classical Review, Vol. 10, No. 9 (Dec., 1896), pp. 424-427,
[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. However, 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.
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).