Should I turn my CD player off when I am finished for the day

Not sure might depend on the player but to save laser hours, turn it off. 

Turn it off and see how is sounds from a cold start.

My CAL ICON MKII has been powered up 7/24 for 20 years (sans system changes/maintenance/power outages) and it's still carrying on and sounding as it should.

I never leave a CD in after play (gathered this info early on).

As Russ notes, this may be dependent on the specific deck.

My deck has gobs of separate power supplies (4-6+?) and I "guessed" that it was better to leave it powered up, plus it takes many hours for it to sound right from a cold start.

DeKay

Follow the makers instructions. If not there then the safe thing to do is turn it off. But one can always do whatever they decide makes them feel best for whatever reason. It’s a free country and Personally this is not something worth worrying about. Worrying is way worse for good sound than most things.

If I were you I’d turn it off and hang it up-side-down in a cool closet. What you use to hang it will effect the sound on subsequent start-up; a metal hanger will make it sound cool and detailed; a wood hanger will be warm and rich. Let it age for a few weeks before restart, then massage well.