Tubes last longer if turned off when not in use. The cathode has a limited lifetime where it is capable of full emission. There may be some turn-on stress, but that is outweighed by burning the life out of a tube.
On turn on, the filament is cold and conducts current with little resistance. The high in-rush current causes very rapid heating and thermal stress. Also, if the unit utilizes solid state rectification, the power applied to the plate come on almost immediately while the cathode may take some time warming up. This could lead to a condition called cathode stripping.
These issues are mostly non-existent if the unit uses tube rectification (the rugged rectifier takes all the abuse and slowly turns on the rest of the unit. In units with solid state rectification, the turn on can be rapid, but a lot of experts think that the "damage" is relatively minor.
It is possible to put a thermistor or other device in the power supply to slowly bring up the power in tube gear if that is a concern.