I wouldn't think there would be any problem either.
Finches are very small and fast birds, so they could easily get away from the chickens if they got too close. You could hang their food at a much higher height than the chicken food, and the chickens would happily clean up any spills or billing out of the seed that finches do.
As far as roosting at night - I wouldn't think them sharing a roost might be a good idea, but finches would need very small roosts - like a tree branch or 1/4" or smaller dowel rods.
Chickens need a much larger roost - like a 2" or larger. Again, I don't think finches would roost on the chicken roost, and if you designed it right, chickens would never try to roost on something as small as at 1/4' branch.