No. There are benefits in turning it off – it offers some protection from voltage spikes and from premature aging of the circuitry, and saves electricity. It may offer protection from someone messing with the speaker wires and creating a short when you’re not around.
I design electronics. Any decent modern amp or preamp is electrically warmed up in tens of seconds, that is, further changes in temperature don’t make any meaningful difference in its performance – measured or heard.
Really ancient stuff? There could be a kernel of truth. Tubes have heaters that take time to warm up of course. Ancient solid state (like early '70s and before) sometimes had janky bias spreader circuits that weren’t well compensated for temperature.
In newer solid state gear, the output stage bias current should be within 20% of the final level within tens of seconds after power on. There’s enough feedback around the output stage that you’re not going to hear any difference while it stabilizes, and it stabilizes pretty fast, much faster than the temperature of the heatsink settles at its final level.
Don’t worry about warm up. Worry about what happens to your powered up gear when you’re asleep or away.
Maybe? There certainly could also be longevity/reliability benefits of reducing power cycles. Mostly for the power switch I would guess. (Thank God for the triac mod that saves the power switches on vintage gear.)
I’m mostly saying there are no sonic benefits to leaving it on.