I don't think there's any chance Dumbledore would have, unless Harry did something completely out of line that put other students in danger. Apart from him knowing that Harry would be returning to a bad life with the Dursleys if he weren't attending Hogwarts (the Weasleys could potentially take him in, but that would break Lily's protection of him prematurely) Dumbledore knew of the prophecy concerning Harry and Voldemort, and that Harry was thus key to defeating Voldemort. Denying him an education in magic and casting him out into the world would have virtually ensured Voldemort's victory upon returning.
Snape, in a particularly emotional state, seeing in Harry only James and not Lily, might have tried to expel him. Dumbledore didn't tell Snape of the full prophecy, he only knows what he overheard the night that Trelawney made it. However, even if Harry was in Snape's house or Snape had the power to expel him, I think it's safe to say that the Headmaster can overrule a Head of House in a decision such as this, and that if Dumbledore couldn't convince Snape to back down, he could still just put his foot down and say, no, we aren't expelling him.
edit: Meant to add in that at the time, Harry is 12 years old, and has only been at Hogwarts for one year. Despite the fact that he's already had an encounter with Voldemort, he doesn't really understand how important he is to fighting Voldemort in the future, and I would guess that he is only just starting to grasp how important he already has been. If you tell a 12-year-old kid that you're going to kick him out of the only place he's been happy in the world as quickly as he discovered it, it's going to spook him and get him to be (a little) more in line.