Can using a bad tube hurt an amplifier?

A bad tube can definitely damage an amp or preamp.

and speakers....they can damage those too. I always use a broom handle when I'm turning on my tube amps...

Yes if you're lucky just a blown fuse and possibly blown resistor but as stated above things can get much worse.

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 .

Sounds like voltage changes if they occur from a tube should be corrected by the auto bias?