A question I suspect reveals a lot of ignorance...

There are 2 parts of the equation. The amount of power dissipated by the plate is determined by the class of operation as stated above. Class A amps like SETs consume maximum at idle and push pull amps minimum at idle.

However, in all case the filament and cathode is sitting there cooking away so even a push pull amp tube will eventually burn out even if you never pass a signal through it.

Don't tubes draw the same amount of current at all times when they are biased correctly?

No

whether or not it is ok to immediately play the amp after switching it on

It won't hurt it. It just won't sound right.