Gandalf is totally an INTP.
"A wizard is never late, nor is he early, he arrives precisely when he means to."
That sounds EXACTLY like the kind of smartass thing I would say to justify/rationalize being late. And the fact that he's late at all suggests Pness. I have never met an INTJ who shows up late. Then there's the fact that Gandalf just kind of has the dwarves take over Bilbo's house without asking or showing any sort of respect for his time or his private property, and he basically laughs about it and tells Bilbo to chill out. He also never really has a concrete, detailed plan and excels at dealing with things on the fly. He generally carries a pretty laid-back, chill, reflective, trollish attiude, smoking and lighting off fireworks.
The only times he gets pissed are when people are behaving stupidly/irrationally. And he often retreats to himself, showing introversion:
"I'm going to seek the company of the only one around here who has any sense - myself!"
And then there's this...
“So do all who live to see such times. But that is not for them to decide. All we have to decide is what to do with the time that is given us.â€
...Which sounds way more P than J. As does this:
"Many that live deserve death. And some that die deserve life. Can you give it to them? Then do not be too eager to deal out death in judgement. For even the very wise cannot see all ends."
Oh, and the clincher is this quote, which is literally the most INTP thing ever said:
"What do you mean? Do you wish me a good morning, or mean that it is a good morning whether I want it or not; or that you feel good this morning; or that it is a morning to be good on?"
I'm not sure where this "Gandalf is an INTJ" nonsense came from, other than that it seems to be rooted entirely in the ridiculous idea that INTPs cannot be good leaders.
This guy was an INTP, and he is widely considered to be the greatest leader in US History and one of the greatest leaders in world history. The point being that Ps can be just as if not more effective leaders than Js. What this idea does NOT seem to be rooted in, however, is anything Gandalf actually said or did indicating he was an INTJ, probably because there is nothing. I'll tell you what wizard in LOTR was an INTJ: Saruman. And their differences are obvious in the way they interacted in the first Hobbit movie, with a skeptical Saruman dismissing Gandalf's theories because he didn't feel the evidence was strong enough to consider Sauron's return to be a legitimate possibility, while Gandalf didn't even bother to listen to Saruman's lecture. And the differences are all the more striking in the LOTR trilogy. They are not the same type.