I think Robusto is right that it is related to stress. The stress change causes the unstressed vowel after the prefix to become reduced.
Two stress patterns exist for the prefix in-
You say that "in general, the prefix "in" negates an adjective, but does not change the pronunciation." I would agree with that, but note that normally the prefix in- is also unstressed, even when it is the third syllable from the end : look at the word infertile /ɪnˈfɜrtəl/, which is the negative counterpart to fertile /ˈfɜrtəl/.
So, I think we have to differentiate two versions of the prefix: the more common and productive version in1- does not take stress and does not change the pronunciation; but there is another, rarer prefix in2- that does take stress and may cause other changes in pronunciation. This distinction also applies to some other prefixes, by the way, such as re-, de- and pre (compare definite and deform). Obviously, both versions of the prefix have very similar meanings, and in some cases different speakers may use different versions. But I have not heard of anyone saying /ɪnˈfaɪnaɪt/, so I think this particular word only exists with the prefix in2-.
Here are some more words with in2-:
Some of these are valid words without the prefix, others are not.
Comparing words with and without the prefix, we can see the kind of sound changes it causes:
- famous /ˈfeɪməs/, infamous /ˈɪnfəməs/
- potent /ˈpoʊtənt/, impotent /ɪmpətənt/
- pious /ˈpaɪəs/, impious /ˈɪmpiəs/
The prefix does not only receive stress, it converts the following syllable into a fully unstressed syllable. In English, vowels in unstressed syllables are reduced.
So that explains the difference in the pronunciation of "fin".
The failure of "silent e" after some unstressed syllables
I'm still not sure exactly why there is a difference in the pronunciation of the final syllables of finite and infinite. I haven't found any other pairs of related words like this that alter the pronunciation of the vowel in the last syllable.
There are a few other words that also show a short unstressed "i" before a silent e like this:
It may be related to having stress on the third-to-last (antepenultimate) syllable. All of these words have this stress pattern, or at least they did originally: nowadays exquisite is often stressed on penultimate.
There are also numerous adjectives and nouns ending in -ate pronounced as /ət/, e.g. associate, delicate, desolate, deliberate, delegate, desperate.