Can using a bad tube hurt an amplifier?

It can cause serious damage ,  I wouldn't risk using a tube if zi suspected it was bad. 

Tube amps will use either an automatic  bias scheme where the amp will maintain the bias voltage for the tube or is is manually biased periodically by means of an LED , built in meter or multimeter.   

If you suspect it's a bad  tube, replace it.    Usually a bad tube will just blow a fuse or take out a bias resistor  ut it can cause catastrophic damage. 

Anyone who has ever seen a tube arc internally knows it's a pretty scary fireworks show. 

If the bias runs away you'll often see the plates turn a red orange , "Red Plating"     Definitely get an amp checked out if this happens with an auto bias amp or if it happens on a manual bias amp that won't hold its bias.   Lethal voltages inside,  have someone familiar with tube amps check it if it's not as simple as bad tube .