I am getting a lot of whining coming from my GPU when displaying static 2D imagery (particularly the darker end of the spectrum). Can someone tell me what the cause for this is and, more importantly, how to make it stop? Is it damaging long-term to the GPU? The card in question is an HIS Radeon HD 5870. Thank you.

It could be the fans... and a lot of dust. Have you cleaned the heatsink and fans with compressed air lately?

a whine like it's beeping? or a fan type whine?

its not whineing for nothing ,so yeah most likely its dust or something else that will damage it if you dont do anything about it. if it isnt already something broken.

Can you tell the difference between fan noise, and a coil?

Fan noise means the cooler is working too hard, or the fan bearings are worn: you will need to clean or replace the cooler.

Coil noise means a coil is oscillating at an audible frequency, and the potting compound that is supposed to dampen this has failed or was applied improperly. It's not a do-it-yourself fix.

See: http://www.tomshardware.com/forum/20853-9-what-coil-noise
and a "for dummies" version: http://en.wikipedia.org/wiki/Coil_Noise

If you get a whining noise only when the the card isn't taxed (as in 2D imagery), then the most likely explanation is that the fan isn't working (smoothly). Dust is a probable culprit. It's possible that the fan still turns when moved with full power (e.g. in 3d games), but gets stuck when it ought to turn slower. Get the card out, remove any dust from it (vacuum it out of the fan), and try again. Turn the fan manually and check whether the bearing is still okay.

I have cleaned the fan and checked the bearing spin, both seem fine. I've had fans go bad before and this doesn't sound quite the same. Hmm, I am going to look into this, perhaps try to find a YouTube representation of the coil noise. Would be a shame if this was it though, since I can't DIY-fix it.

EDIT: Appears people get this with my PSU as well (Corsair TX650W), still looking...

Some graphics cards emit a high-pitched noise when rendering at very high framerates.. it's possible you might be able to fix the issue by forcing V-sync in whichever application you are running.

If it happens whenever any dark image is on-screen, it might actually have something to do with your monitor - my old screen made a constant high-pitched ringing sound whenever the backlight auto-dimmed (after # minutes without input or in dark scenes) and when it was on stand-by mode.
If you're using a VGA cable to your monitor it's possible that one of the timing settings is set incorrectly - auto-adjusting your monitor or manually correcting the setting could fix the issue.

my graphic card was also making more noise then usual at one point. it was a geforce g210, in which you couldnt get to the fan very easily if not impossible.
http://www.prijsvergelijk.nl/images/prodhigh/3844667-HP.jpg

eventually i went with my dad to the computer store, he needed a new joystick and i saw a geforce g430 . we got a 30 euro discount. basically making the card cost 30 euro.
its alot more quiet again.

I was just about to suggest this. ;) I never jeard of coil noise before, but after reading cjrgreen's post, I searched youtube for "coil whine" and "coil noise". There are many clips, but the actual sounds vary quite a bit. But perhaps you can find one that's close to the noise you're hearing.

I notice that as well (the differentiation in coil whine, I mean). I remember reading a long time ago that you can put nail polish remover on the coils and that will lowerthe noise, can anyone attest to this?

Check this video and fix your annoying GPU Whine

http://youtu.be/eqf4xQ3eL0k