I can't think of a reason why playing while the tubes warm up would hurt them, except it heats them up faster. With anything (especially vacuum tubes with all their precisely-made little bits inside) it's always better to let them warm up slowly. All those bits expand and contract at slightly different rates, so it's best not to rush them.
More important is the question of when the high voltage (B+) gets applied to tubes. Putting B+ on a cold tube (before the filament has the guts warmed up) causes something called cathode stripping, which will shorten the life of the tube.
This is why amps have standby switches - in standby, B+ is shut off. So, you let the tubes warm up, then switch from standby to play. That way, no B+ on cold tubes. This really only effects amps with solid-state rectifiers, since they supply current the instant power is applied.
In amps with tube rectifiers, the recifier tube has to heat up before B+ can flow. THis naturally delays the high voltage, so the other tubes have a chance to warm up, too. So with tube rectos, a standby switch is merely a convenience.
I don't think waiting between standby and off buys you much. More important is letting the amp sit and cool off before you move it. The shock of moving hot tubes (especially if you move it outside into freezing temperatures) is rough on them.
