Which one is grammatically correct?
I don't know when he leaves for New York tomorrow.
I don't know when he will leave for New York tomorrow
Is it correct to use the future form of the verb leave and the word tomorrow in one sentence?
Both are grammatically correct.
Yes, you can use such words as tomorrow, this evening, later today, next month, five years from now in sentences that refer to future time.
But there is no "future form" in English. The simple present tense form of to leave is leave/leaves (he leaves); the simple past tense form is left (he left); the simple future tense form is ??????. English does not have one.
If there were a future tense form of leave we could just write it, something like leavek (he leavek).
To say will leave is the future (tense) form is to arbitrarily choose one of over a dozen ways to refer to future time as "the future form".
Actually, the modal will expresses current resolve:
I will leave for New York tomorrow
expresses the current resolve of the speaker to leave for New York tomorrow. As such, it is a promise. Even such statements as The sun will rise tomorrow and That kind of thing will never happen here are promises.
I don't know when he'll (he will) leave tomorrow
expresses the speaker's ignorance of the exact time tomorrow that the person he will fulfill his resolve to leave. He has made his resolve known, but perhaps he has not stated at what time tomorrow he will carry it out.
We use the simple present to talk about future time with the strongest certainty. Things we are absolutely convinced to happen:
The sun rises tomorrow at 0626.
The train arrives tomorrow.
He leaves for New York tomorrow is the way to say with the strongest certainty that in the future time designated 'tomorrow', the person he => /leave/ for New York.
I don't know when he leaves tomorrow
expresses your ignorance of the exact time that his leaving happens. You know he's leaving tomorrow, just like you know the sun's rising tomorrow, but you don't know when tomorrow or at what time tomorrow.
How else can we refer to future time? Here are some ways:
He's leaving for New York tomorrow.
He's going to leave for New York tomorrow.
He's going to be leaving for New York tomorrow.
He'll be leaving for New York tomorrow.
He is to leave for New York tomorrow.
There are others.
In its simplest state, the difference between he will leave and he leaves is in the certainty, which in turn dependes on how far along the process of achieving a target the speaker is. This runs from will as an idea or intention, to using present simple for a fact or repeated action.
In more detail, we use will initially when it is an intention but nothing concrete has been done. Will has the complication of being modal ('varies by mood', eg subjective) as an intention can be weak, strong, or anywhere in between, but for stating the future "I think I will fly to Thailand" is just an idea. Intenions beget plans, such as looking at flight prices and schedules, when you would start saying "I am going to fly to Thailand soon", as going to requires evidence or reason. When you start booking hotels and flights, it becomes a continuing action in progress so you could use the continuous and say "I am flying to Thailand soon". When you have your ticket, and your seat, and a time and date you can put on your timetable so it is as agood as factual, you could use the present simple and say "I fly to Thailand tomorrow". Present simple can also be used for timetabled events ("the flight leaves at 9:30am every morning") as it is a repeated action.
In summary, he leaves and he will leave are both grammatically correct, but he leaves signals that the speaker is a lot more certain that this future event will happen.
This apparent complication, by the way, is because the future can never be spoken about with the same certainty as past or present events, so the system evolved to indicate the strength of belief held by the speaker.