I've got a older monitor (it's most likely pre-2000, or close to) and when I first power it on, it fragments. The cpu information and windows version screens turn choppy, and seem to move up the screen. It also emits a high pitched whistle. Both problems disappear when:
A:I turn the screen power switch off, then on,
Or B: As soon as the GUI desktop first appears.
What's the problem?

P.S. When I shut down my computer, I leave the screen power switch on and usually flip the 6-plug outlet I'm using off. (Yes, silly, I shut down the CPU first.) I've tried turning the screen switch off, and it doesn't change.

Thanks for any help you can give!

The screen has what's known as a "dried cap". That means that 1 or more of the electrolytic can type capacitors has died. You can spot them fairly easily if you feel up to taking it apart as the tops usually dome up and/or they leak fluid. Capacitors are cheap to buy and quite easy to replace provided you can solder. Have a google for pictures domed capacitors to get a better idea of what to look for!

Erm... Is it safe to let it be?

That's a difficult one to answer. It might carry on like that for years or it could stop working tomorrow.
They are not particularly difficult to replace though, I have replaced caps in well over 100 lcd screens!

Another question, now that I think about it, is Just how vital is this piece. It's been doing this for quite a while...
And there isn't, say, a fire risk, is there?

The fire risk is minimal. The monitor will be protected by several fuses.

I think Rik has this spot-on. Either replace the monitor (flat panel displays w/ similar or better resolution than your old CRT are pretty cheap today), or roll up your sleeves and replace the caps that are failing, or just keep on like you are until it lets the magic smoke out and you have to fix/replace the unit. Myself, I'd probably opt for the latter, being a lazy sod! :-)