The mums have to eventually stop blooming .. otherwise they won't have any energy to come back next year!!!
Don't panic. If the mums are indoors as houseplants and you have the space - let them stay that way. Clip back the dead blooms (dead-heading, essentially) and water the plants occasionally and make sure they get sun. They should be fine. They may eventually die back some, but this would happen outside every winter anyway - so it shouldn't hurt them.
If they are outside and you can't plant them - just make sure they are protected from a freeze. Wrap the pots in burlap, or foam, or buy the old rose styrofoam cones and put a pot in each one (styrofoam is a great insulator). If the mums are cold hardy (not all are) they will die back, but if you keep the roots from freezing, they will come back in the Spring (little green shoots at the base of the plant mass).
Good luck!! Mums are usually pretty easy ... so unless you have one of the really fragile non-cold tolerant varieties, you should be fine.