Can using a bad tube hurt an amplifier?

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?  

Yes, I just had this happen to me: everything working fine for 4 months with the 300b Gold Lion’s in my Coincident Frankenstein stereo amp. The amp and tubes are both 4 months old, I haven’t touched anything in weeks, and I turn it on a few weeks ago and one 300b tube is much brighter than the other one. By the time I got to turn it off I hear sizzling bacon, I shut it down and witness what appears to be plenty of smoke bellowing from the amp. Israel Blume Coincident owner says the tube has "run away" which I believe is the same as "red plating" as stated above. I have no way to check tubes so I sent if off to the retailer and they have checked it and confirmed it is bad, warrantied it, and a replacement tube is on the way back. There is a 1% failure rate and I am one of the lucky ones.

There is automatic bias circuit in this amp so nothing to adjust on my end, but no special protection circuit to shut off the amp in the case of a faulty tube, not sure how many amps have this? 

No warranty for the amp as it was the tubes fault. Easy fix though thanks to point to point circuit wiring. 

The joys of tube amps I suppose.