I know I can use "doggie" to refer to a dog and that's a diminutive form, but what about words like "selfie" or "foodie" that aren't necessarily meant to diminish by adding an "-ie?" Is there a term for that kind of suffix?
The -y and -ie suffix is not only used for diminutives, but also for pet names (e.g. aunty) and other "cute" words. This last use hasn't made it into most dictionaries, but it's clearly how selfie relates to self. foodie appears to use this sense, and perhaps also some of the sense through which hip relates to hippie.
A more general term for these types of uses is hypocoristic.
It is an alternative spelling of y whose usage dates back to the 16th century:
alternative spelling of -y; now mostly of -y (3), but formerly of others.
Use with common nouns seems to have begun in Scottish with laddie (1546) and become popular in English due to Burns' poems, but the same formation appears to be represented much earlier in baby and puppy.