I dyed and then permed a friend's hair for her in the same afternoon - and no, it didn't fall out or turn a funny colour, but you'd have to do it at your own risk!
Wait a week before perming after coloring hair. If you do both together the perm wont take properly. Also if all done in one day the hair would have been shampooed at least four times and the conditioner would work against the perming process.