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.

I have had my Nakamichi 5 disc on for 20+ years.
(Well, I moved it a few times, and we have power outages.)

The good part about it is that when I do start it up it is a crap shoot whether it will be Bad Religion, Doug McLeod, Immortal Technique, Simon and Garfunkel, or Bach.
Sometime it has a different CD in it, which is a shocker.

I think the question the here is whether the laser is still reading the non playing disc, and if that will therefore decrease the laser life.  This is a non trivial issue.  Lasers can be replaced but it isn’t easy.  Many threads here are about replacing lasers note difficulties due to discontinuation of player models, smaller companies no longer in Buisness, having to ship to Japan for repairs, etc.

  I always assumed that when the disc stops playing, the laser stops reading.  I really have no idea and now I will be more compulsive about turning off the player

If by “No Disc”, you’re saying you might have a Jays Transport, I posed this same question to Alvin at Jays.  He said, “Take the CD out, close the cover, but leave the transport on”.

turn the frigging thing off.

system on, diddle with something else for 20 minutes, good to go.

I’ve never turned any of my CDPs off unless I was going on vacation or there was a severe thunderstorm.  This applied to all my non-tubed electronics, as well.