Tubes are thermally stressed by being cycled off and on and this shortens their life span. They're also stressed (though to a much lesser extent) by simply being on. It's generally considered better to leave a tube amp powered up, even for several hours, than to turn it off and then back on. Most recording studios leave their tube gear on all day and only turn it off at night, and some even leave their solid state gear on perpetually, although they always have sophisticated protections on their regulated power supplies.
The filaments in tubes are similar to incandescent light globes. There are always microfractures in them during manufacturing, and these areas get slightly hotter than the rest of the filament and so it stretches slightly and thins out. Eventually as this gets worse over time with on/off cycles the necked section will get too thin and fail, going open circuit. Filaments have the lowest resistance when cold, and therefore draw more current; that's another reason for failyre at turn on. There is a name for this but it eludes me at the moment.
In some tubes, because of design and/or manufacturing tolerance, suddenly applying the HT to a cold tube can cause enough deformation in the physical structure that there can be a short or breakage caused in an element, therefore failure of the part and sometimes the surrounding circuitry. Do this enough over time, and there will be a failure. There's lots on both these issues in the old military reliability data.
All my hifi tube amps have tons of solidstate in the; to slowly bring the cathodes up to temperature over 2 mins or so, the slowly bring the B+ (HT) up, then the biassing. The tubes get a very gentle time and long life. None of this is hard to do, but to do it well is expensive enough for the modest scale of construction of tube power amps that it would likely add a decent amount to the sale price. Besides, a lot of 'purest', who usually don't have a clue, don't like any SS in amps, even if it's not in the signal path.
My new tube hifi preamp will likely never be turned off, but the poweramps will as sometimes I don't use the system for a couple of days, and it's quad-amped, with all but the sub (<60Hz) tube.