The paradigm of fio is quite unusual - see its present indicative partial paradigm (fīmus and fītis are pretty much regular; from my undergrad textbook, Sobolevskii 1948):
Weiss writes that
"In Classical Latin fīo, fīunt, fīam etc. 'become' have been analogically restored" (p. 126).
It seems this explanation goes back to Sommer 1902 (at least, that's what Leumman 1977 mentions, a later edition though). Here's what Sommer suggested (p. 589):
So, Sommer hypothesized that ī was "restored" in the thematic forms with ī+V so that the paradigm of the verb would have fewer alternating forms. This is known as analogical restoration in historical linguistics.
cf. de Vaan "*fī - would develop phonetically in the 23s. and 12 p.pr. forms, and then spread to the other forms of the paradigm."
Sihler adds, cautiously though, that it could have happened because of a very special status of fio - "the only verb in -io with tonic accent on the i" (p. 546).
Of course, we won't find "hard evidence" - we are talking about Old Latin here, after all. But another source of indirect evidence comes from Plautus and Terence, where we can find both fĭerī (more common) and fīerī, fĭeret (more common) and fīeret etc. See Sihler (p. 546) for some interesting thoughts on this
Weiss also mentions, in footnote 10 on the same page, a very interesting observation made by Jan Safarewicz (Safarewicz 1974): ī tends to be used in those word-forms of fio when the following syllable is heavy. Tronskii 1960 argues that ī > ĭ occurred only before -er-. That being said, Leumann 1977 still writes, "Die Verteilung von fī- und fi- ist unregelmäßig" (p. 530).