Can using a bad tube hurt an amplifier?

@emergingsoul 

Which tube amp do you own or are considering purchasing?

When you mention "and then voltage changes over time gradually" are you referring to tubes going out of bias or actual voltage changes from the power grid?  If the former, it is imperative that a tube amp be easy to bias or self-biasing, which most newer offerings are.  

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 .