"Ei" is almost never a diphthong.
The exact list of examples depends on what you call a "diphthong".
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.
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, 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.
/ej.j/ before a vowel
Some instances of <ei> are thought to have been pronounced /ej.ji/. Cser mentions reicere (p. 149) and the genitive form Pompei (p. 13).
Words with /ej.j/ followed by a vowel other than /i/ have had spelling variants with <ej>, like the word eius/ejus mentioned in Joonas's answer. One thing to keep in mind is that some sources, particularly older dictionaries like L&S, may write this as ēi, where the macron represents syllable weight rather than necessarily representing the length of the vowel itself. Intervocalically, /j/ is almost always found geminate in Latin—that is, as /j.j/, but it became usual to write this with just a single letter I (or in modern-era texts, J).
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).