I haven't been able to find any solid information on the etymology of the Latin infinitive, which is frustrating. I'm sure it's out there, so this will be only a partial answer (containing a decent helping of speculation) for now.
But one thing seems fairly clear: PIE didn't have an infinitive per se. Different language groups developed it independently post-PIE. In Italic in particular, a verbal noun with the suffix -si developed: *dōnā- "give" → *dōnāsi "to give". In Latin, this developed into the active and passive infinitives dōnāre and dōnārī through rhotacism (and presumably an extra suffix on the passive, though I can't find any details on this part—but final short *i > e is regular).
In what became the third and fourth conjugations, there were three different types of verbs: the consonant-stem thematic verbs, the *i-stem thematic verbs, and the *y-suffix verbs. The first of these had infinitives like *edesi (from *ed- "eat"), the second had infinitives like *θakiesi (from *θaki- "make"), and the last had infinitives like *gwenyesi (from *gweny- "come").
But the verb that would become fiō was special. Its stem was originally *fui- from PIE *bʰuH- "become, come into being", and this led to an infinitive *fuiesi.
Latin didn't like having sequences of short vowels together. So the *u ended up disappearing from the present forms (but not the perfect, which became Latin fuī). But this stopped the ending *-iesi from contracting into * -esi, which happened in the other third-conjugation i-stems (* θakiesi > * θakesi > facere). The vowel deletion only applied once, and since it deleted the u, it never went back and deleted the i.
This left infinitives *fiesi > *fiere, fierī, with an extra i before the ending. And this i then prevented the passive from contracting into *fiī as happened in the rest of the third conjugation.
So fierī looks special simply because of an accident: it had an extra vowel that later disappeared. As for why it uses only the passive infinitive, I really can't say ("it's semi-deponent" is a cop-out).