The fairly obvious:
Latin orthography is incapable of distinguishing /i/ from /j/, /jː/ (eius /ˈejːus/), and /ji/ (adicio /adˈjikioː/). These are phonemic distinctions: <i> between vowels is almost always /jː/, but it's /j/ in e.g. Gāius; <i> between a not-vowel (word boundary or consonant) and a vowel other than another <i> is almost always /j/, except in loans like iambus /iˈambus/; /ji/ isn't marked at all, because it doesn't "naturally" occur anywhere except in zero-grade compounds of iaciō (e.g. adiciō /adˈjikioː/), where the /j/ is sometimes restored by analogy (but actually adiciō /adˈikioː/ is also attested).
(Conversely, I expect but have not confirmed the distinction between <ī> and <ii> is spurious at least some of the time.)
/w/ and /u/ are also certainly separate phonemes in Latin: there's a minimal pair in volvit /ˈwolwit/ 'he rolls' and voluit /ˈwoluit/ 'he wanted', spelled the same.
Also, the diphthongs ae /aj/, eu /ew/, oe /oj/, au /aw/, and ui /uj/ are not distinguishable from the corresponding sequences of two vowels: cui /cuj/ vs. fruitur /fru.i.tur/. Eu as a diphthong exclusively shows up in Greek loans and the interjection (ē)heu; in native words, possible confusion is limited almost entirely to a few instances of ui in practice.
The Greek phonemes that got their own spelling (ph, th, ch, and y) were certainly not separate phonemes for most Latin speakers. Likewise, pulcher was surely /'pulker/, not /ˈpulkʰer/, for almost everyone.
For a more out-there tack, a case could be made that vowel nasalisation, which was not reflected in writing, might have been phonemic for educated speakers:
The well-known instance of nasalisation is before word-final -m (mālum [ˈmaːlũː]), which is perfectly regular so you don't really need to represent it in writing: it's allophonic. It also happens regularly before a nasal consonant followed by /s/, and this is where it possibly breaks down:
E.g. consul [cõːsul] is frequently written as cosul, which is a non-phonemic spelling but can be dismissed as just an error; certainly the spelling consul also persists. It persists so hard, even, that eventually the n is restored to the pronunciation of the word, which is now [cõːnsul], still written consul: the nasalisation has become phonemic, and it is not represented in the spelling.
This is admittedly not an uncontroversial statement: all we know for sure is that in common speech, the n in these positions was not pronounced and the vowel was lengthened compensatorically (this is what we see reflected in the Romance languages, and also mentioned explicitly and implicitly by contemporary grammarians), and that educated Romans did tend to pronounce the n but also keep the vowel long (stated explicitly by Cicero and various grammarians). As far as I know the nasalisation is imputed entirely on typological grounds (Allen does not specify, and I'm away from my other books at the moment; the Romance languages all lost nasalisation everywhere so it's impossible to tell based on the descendants), but eminently plausible; that this nasalisation would be preserved in educated speech does not strike me as unlikely (especially since even the likes of Cicero neglected to pronounce the n in some words, surely with nasalisation there), but isn't directly confirmed.
It is, at least, a thing to consider.