Lily: INFJ. ?? Why an IN? From the little Lily scenes in DH, she proved to be quite a bit judgmental(unknown to her), making hasty judgments, being impulsive. She didn't appear to have a gift for reading people, or guessing their motives. She is the black-and-white self-righteous person and certainly isn't empathetic. Yes, she was concerned with morals, and wanted to change the world for the best(probable reason for joining the Order) but didn't act very fair with everyboy. (Didn't give equal chances...). She is quite unforgiving too. She's an obvious flawed J, and even an FJ, since she appeared to follow her heart more than her head(went out with James even though he was a bully and she hated bullies, (but also she thought he had changed and didn't know James still bullied behind her back either), bashed Severus's friends but made excuses for the Marauders who were equally crappy people at the time, bullying everyone "beneath them").
It is clear that Lily didn't follow any reason or logic, thus making her an irrational F. But honestly, did we even see Lily using N(Ni, Ne) or S(Se, si)? I don't think so, unless something escapes me, so how can we classify her an N or S? As for I vs E... Well, Lily, according to JKR, was quite popular, Giiny-alike(don't agree, but whatever), sociable, had many friends. It can point towards a E preference. But, she remained Severus's best friend for approximatively seven years, and tried to keep Petunia in her life, showing loyalty and preferance for having a few close sincere friendships. That might point to an I preference. Obviously, Lily ended up being disloyal to Severus(but to her defense, she probably didn't notice it, and it was provoked by her lack of understanding, Severus's temper, and serious misunderstanding and bad communication between the two) afterwards. And, the fact that adults were all about James's friends, never mentioning Lily's other friends than Severus can be a proof that unlike her friendship with Severus and Petunia, which didn't work out due that Petunia ditched her and that she ditched Severus, her other relationship may have been shallow superficial ones(points to E preference).
There are many contradictions showing Lily as an E, as an I... But I think there's more indications of her being an E.