I was struggling with something like this a while ago, about the most general solution is to test all the line segments of one object against all the line segments of the other. That'll give you a few segments that intersect, and if you're lucky one of them will be a tile edge. If you're unlucky you'll get two tile edges when you bump into a corner. If you're really unlucky you'll get three edges when you totally swarm over an edge.
You could try seperating all the cases, but that could be tricky. A more mathematical way would the method of separating axies, which gives you a separation plane and penetration distance. Google should have quite a bit on it.
If you're sure one will always be static (like with tiles) and all your tiles are the same shape then you can probably simplifiy things quite a lot. For a start you should be able to use your velocity to trim down the possible edges to one or two (can't bump into an edge you're moving away from
).
