The biggest issue is likely to be background network use. For example, Amazon's Kindles will continue to use Whispernet and/or WiFi in the background. Assuming you have turned those features off, you would want to power it off entirely only if:
You plan to store it for a long period, on the order of months.
You plan to open up the physical case for some ...
The first thing you should always do is look in your devices manual if there is a recommended way of cleaning (or doing anything else). It is probably the best way to avoid problems with your warranty (in case something goes wrong).
If you don't have your PRS-650 manual any more you can find it (googling sony prs 650 support) on-line on the Manual, Specs ...
After a little research, I came across this post. It details two different ways to clean eReader screens. One for E-Ink screens and one for LCD screens.
The Coles notes version for E-Ink screens:
Use a dry cloth or dampen a cloth with water (microfiber preferred).
Do not apply water directly.
Carefully wipe screen with cloth and allow to dry before using.
Over time, grease and grime from fingers transfers to the display and can cause touch sensitivity to become erratic. Simply clean the Nook display with a water-dampened cloth and touch sensitivity will return to normal.