It can cause damage to the tube's cathode coating. It is called cathode stripping. A problem mostly to power tubes running from plate supplies of 400v or greater. IMO, new current production power tubes are more prone to this than NOS.

>Do you mean that the tube rectifier wastes amplifier power or electricity?

It wastes electricity, adds heat to the system (never a good thing), and represents an additional stress on a power transformer.

>Can you wire the standby switch to become completely removed from the circuit after warm-up?

No. Why would you want that?