Can it?

With the snow as of late I am getting snow accumulation in my rims. It is causing the tires to be out of balance pretty bad. Bad enough that I avoided the highway this AM because of severe stearing wheel shake at speed. Under 35/40 was not too big of a deal.

This afternoon I need to go to a coin operated place and wash them out...

 

As far as the vehicle is concerned, such extreme imbalance is rough on the shocks and the rest of the suspension, it's particularly hard on the steering system causing it to run in it's 'boosted' mode continuously. On the safety side of things, the imbalance causes reduced grip and less precise steering feel.

While slow, short drives in such conditions would not cause any sort of damage to the car, extended high speed use is definitely not a good idea. Poke the ice and snow out of your wheels whenever possible.