How does the Latin “SAPIAT” become “sache” [saʃ] in modern French?
I thought the reason may be palatalisation as [ʃ] seems to be a palatal sound, but I can't think of any other examples to prove this theory.
TL;DR: indeed, it is a kind of palatalisation, other examples include cavea > cage, rabia > rage, jam > déjà…
As usual in this case, I recommend the excellent Petit précis de phonétique historique by Noëlle Laborderie. In this case, sapiam's evolution is studied in chap. 16 3.2. I will make a short summary adapted to the case of sapiam. I will use IPA so, latinists, feel free to add your filthy transcriptions in Bourciez' alphabet if you feel such urge.
Here¹, palatalisation, means “a shift of the place of articulation toward the hard palate” which can result in a sound shift, for example the palatalisation of the [k] sound in Latin cabalus yields a [kʲ], then a [tʲ] and eventually a [tʃ] which lead to the [ʃ] in French cheval.
There are two main triggers of palatalisation in the study of the evolution of Latin into French, which are: reinforcement of an already palatal sound and assimilation with a contiguous palatal sound (usually the palatal approximant [j] or a front vowel).
In Laborderie's terminology, inherited from Georges Straka, a palatalisation is said to be true if it is due to both of these phenomenon and fake if not.
The appearance of [j] in intervocalic positions
The accent of Classical Latin has affected most of the subsequent phonetic evolutions of Latin, without further details, let us admit that the accent in Classical Latin sapiam ['sapiam] was on ['sa], that the [i] was a so-called breve vowel and that all breve [i] and [e] evolved in [j] around the first century BCE, so with the unrelated loss of the final [m] it became [sapja]
The (fake) palatalisation of [j]
The palatalisation of [j] is a fake one according to what we saw: since [j] is already palatal, what happened was only a reinforcement.
This kind of palatalisations happened around the III century CE and its result is in general an affricate, generally [d͡ʒ ]. But in this case, the [j] is right after a bilabial consonant, [p], which altered the palatalisation, due to the non-existent role of the tongue in its articulation. So instead of [d͡ʒ ], it yielded [ʈʲ], which then yielded [tʲʃ], [tʃ] (around the VII century CE) and finally [ʃ] around the XIII century CE.
You asked if there was other examples of this phenomenon. If found no other example of this precise occurrence [pj] > [tʃ]², but there are several other examples of fake palatalisations of [j], both in initial position, where the [j] was already present in Classical Latin such as iam > déjà or diurnu > jour and in intervocalic position, where it comes from a mutation of a [i] or [e] in hiatus, such as rabia > rage or cavea > cage.
¹ as everywhere else as far as I know.