I know this is a pretty old topic, but I'm having the exact same problem with my MSI 7870, but I've gone out and bought a PCI-E 1X sound card (Asus Xonar DSG 5.1) and still have the problem, no matter what I used to play sound I got the interference whenever the card ramped up, did seem like it got higher pitched as the fps went up, and most people say to use V-Sync but I get horrible lag input in almost every game I use vsync on :/
I've also swapped out everything as the MSI's tech support told me to do, Motherboard got rma'd, then upgraded from an MSI z77a-GD45 to a Z77A-GD65, still there. RMA'd CPU, then tried a different PSU as well (had a Corsair HX850, tried my PC Power & Cooling 750 Watt Silencer PSU, still there), tried the card in my wife's computer, problem was there. Tried my Geforce 9800GTX+OC in the Computer and guess what? NO NOISE!!
Called up msi, and first call the guy told me to try a different video card (this was before trying 9800) and then immediately hung up on me, never took my name or anything. 2nd call, guy told me to underclock the card because all of the ghz chips can't handle any oc (which mine is a 50mhz oc FROM MSI), tried that and it's still there. Called a 3rd time and the guy flat out told me "Yeah I'd avoid AMD for the time being, they're not making anthing good at all right now." And then I asked what I could do, "well you can exchange it for another one and probably continue to have the problem" and I asked if there was anything else they could do like swap for an Nvidia equivalent, he said he'd have to get it approve and he'd call back that night or the next morning, ok hung up and waited, then by 6pm the next night I finally called back and got back to the person, "Oh yeah." was his reply..... Then he finally tells me, we can exchange it for a 660 (non-ti, hence slower than 7870....) And then I asked could I do something better and I'll pay the difference? and he replied with "our company doesn't do something like that" and basically said take it or leave it and tough luck. considering I've only had the card for 3 months and paid $240 for it, sure it's gone down 30 bucks but I told them I'd pay the difference for a better card on it's current value not on what I paid and they still blew me off.... Definitely will be avoiding MSI from now on, which stinks because I've been getting Motherboards/Video Cards from them for over a decade, this has been my only technical problem I haven't been able to solve on my own, and first time having to contact them and then find out they treat their customers like garbage is horrible.
Also I even tried putting ferrite chokes on my sound cable and it did nothing... And also the whine from the speakers matches the whine from the video card itself, and like I said, installing the Geforce 9800GTX+OC in there the noise completely went a way, Klipsch Speakers, Headphones, and Monitor speakers were all silent on that card...