We sort of have the same thing in English with the word "with". Usually it means in cooperation, but sometimes it means "against", as in "fight with", "argue with", etc.
That would be jag springer till huset. I mean, they're closely related and I'm sure there are many situations where you could use either - but there's still a difference: either you're running in the direction of the house, or you're running to end up at the house.
Would till huset cover the situation of running into the house? (Either in a door or crashing into the side of the house?)
No, we'd use in i for that, rather than till. That goes for both entering and smashing into the house.
Would "I run against the house" be an appropiate, albeit unlikely translation?
Yes, and that's funny, I was just running away from a house in a video game not ten minutes ago.
I don't think it should be an accepted translation, though. :)