The questions is self-explanatory. I've actually seen "lean on door" to be more frequently used, but I've also heard the latter form.
Is there a difference between these two forms?
lean against (transitive lean) =stack, position.
leaning the pictures against the door to see them better.
slammed the door shut and leant /leaned his back against it.
cf leaned the ladder against the wall (free dictionary)
lean on =use force against
leant on the door until it opened
he had to lean on the door to get it to shut properly.
There is considerable overlap in usage, but 'lean something on the door' is not found.
As you can see from this link the meaning is the same and they can be used interchangeably to refer to a physical posture.
I think the real difference is that lean on can be used also figuratively in the sense of " having as moral support":
Referring to the two sentences you are suggesting: lean against vs on the door, according to Google Books lean against is more common.