Strictly, if someone kicks or uses their shoulder to open a door, so that it is thrown open by the force, then we would say 'lick in'. If in the process of ramming the door with one's shoulder the whole door comes off its hinges and falls to the ground, then it's 'licked down'.
In everyday speech, 'kicked down the door' covers whether it was 'in' or 'down', and whether they actually kicked it or used their shoulder.
Again, to be precise, we don't actually 'kick' the door, as in the way we kick a football. Boy, that would hurt. Police, or the bad guys, slam the heel of the foot into the door near the lock area.