So I have a Marshall DSL40C 40Watt amplifier and there is a "Stand By" button on it.

Can I leave my amp on Standby all the time or will it burn something out if I leave it on for days, weeks, months or years?

Sorry if this is a stupid question, but I would like to know.
Why would you do that? It's one switch, what benefit would you expect from wasting power and leaving it on?

From a technical standpoint, there is current going to the tube heaters when the amp is in standby. This is fairly negligible from a tube wear standpoint, but keeping the tubes and components hot when they're not in use is probably not a good idea.
So no, you probably won't burn anything out right away by leaving it on for a while, but you're going to wear out the components for no good reason.