Leaving the heater of a vacuum tube on all the time vs cycling it
A heater's failure mode is typically a stress-related fracture of the tungsten wire or at a weld point and usually occurs after many thermal on/off cycles.  One way to mitigate this of course is not to turn off the heaters at all (one of your options).  Another is to employ a negative temperature coefficient (NTC) device such as a thermistor in the power supply servicing the heaters. to allow the heaters to reach operating temperature more gradually. 

When the ENIAC computer was built in 1946 using over 17,000 vacuum tubes, the failure rate was initially several tubes a day.  Of course they were already on all the time.  They derated the voltage (and current) going to the heaters and reduced the failure rate to one tube every two days (longest time recorded without failure was five days).

Leaving tubes on all the time can accelerate failures which occur over long periods of time (thousands of hours of operation).  Cathode depletion is the loss of emission after thousands of hours of normal use, as it is poisoned by atoms from other elements in the tube.  However, according to page 34 of the 1960's era book getting the most out of Vacuum Tubes, this is fairly rare since by the time the cathode has lost its emission, the tube is pretty much dead for other reasons.

This same book, on page 14, makes another suggestion re keeping the heaters on; during standby, reduce their voltage to half instead of either leaving them on full voltage or turning them off.