Can you leave class D amps on and not ever turn them off?

You SHOULD keep class D amp on 24/7.

A reliable surge protector might not help if lightning strikes, so if that's the case of such vulnerability, than turn on and worm up for 20min and than enjoy the tunes.

Class D amps as no other amps can be on stand-by forever. It will make their life longer than if they're completely turned off for months.

Do what maker recommends.   Bel canto recommends my amps be left on so I do normally except if away for more than a day.   Class D should draw minimal power when idling and does not generate much if any heat due to efficiency so that makes it more practical in general.

My Triton's have a class D amp (bass amp) and there is no power switch on the speaker. The manufacturer (Golden Ear) merely states that when an audio signal is present the amp is "on" and when no audio signal is present for a period of time (20 minutes or there about) the amp goes into "stand-by". However, even in standby mode they do draw power (albeit very little) and for that reason and other piece of mind reasons when I'm not listening I power them down via a switched power conditioner I have them connected to. I only turn them off when they have gone into standby, not while the amp is still actively on (as evidenced by an LED). 

Thanks for all the responses.  
They are a little mixed.
The concensus seems to be to leave them on.
I've in the past always turned them on and off, and have had trouble with one balking at startup.

FWIW, the BEl Canto ref1000m amps draw a lot of current when powered up.    If I power both up at exactly the same time, my circuit breaker will break.   Once on though there is never an issue.

So a major power draw event like that is likely best kept to a minimum and is an indicator to me that its best to leave them on if possible.