In the real world, as it operated for many decades, ac operated indirectly heated heaters in valves (tubes) was the overwhelming standard.
They were used in low signal level ampliers with no ill effects.
Occasionally, a piece of test equipment would use DC for the heaters, but, in that case for signals several orders of magnitude lower than any audio or RF equipment.
Portable equipment usually used directly heated filament type cathodes, because they were more efficient emitters, & the designers were struggling to get the most performance possible for the longest battery life.
One place where indirectly heated tubes were widely used with DC heaters was in car radios.
The use of DC was obviously because it was available at a voltage useable with 6.3v heaters, indirectly heated tubes were used because they offered higher performance, & also because the "DC" from cars is quite noisy
This latter was likely to be injected via the directly heated cathode & lift the noise in the signal passing through the amplifier stage.
One of the standard tests for indirectly heated tubes was "heater to cathode leakage ".
If this was out of spec, tubes were discarded, & replaced with new ones in Broadcast, Comms & other critical work.
Some, if not many tubes available today are "pulls" from years ago, which have been tidied up, (maybe) tested for emission, put in new boxes & sold.
If these have out of spec' heater/cathode leakage, (which many do), & are put into audio amplifiers, they will almost certainly cause hum.
If the heaters are fed with nice regulated DC, such tubes are probably quite useable, so the urban myth has developed about running tubes with DC heaters.
Lifetime/durability?
With new, in spec tubes, the lifetime with ac heaters was normally many thousands of hours for the less stressed ones, rectifiers & power output tubes had shorter life expectancy.
Car radio tubes seemed to have similar lifetimes to those in domestic radios, which was a bit worse to a lot worse than those used in Broadcasting/Comms.
In conclusion, I would say there is no difference, between ac & DC, all the cathode sees is heat.
Open circuit heaters are fairly rare, most cathode faults are when the cathode material loses emission, or the insulation between heater & cathode fails.