My Lumia 1520 is about 6 months old and from the beginning, had the well-known issue where swiping around the bottom third of the screen wouldn't always register or it would misinterpret as a tap and open whatever my finger was on. It was rare though and didn't bother me much. About 1-2 weeks ago though, I started experiencing what you are reporting. It first happened on my Start Screen, where it would spaz up and down a quarter inch or so, super-fast, and usually locking the phone and unlocking it would resolve it, but it would sometimes be back within a few seconds. Sometimes, it'll even do it at the lock screen, sometimes "tapping" as well, so I'd end up locked out of my phone for a minute because of too many "guesses." It really got bad after I upgraded to Windows Phone 8.1 w/ Lumia Cyan, but I think it is just coincidence. I know there has been some tweaking regarding the "no swipe" or "random tapping" at the bottom-third of the screen, so it's _possible_ the Lumia Cyan update is responsible, but I highly doubt it. (Prior to about one week ago, I was running the Developer Preview w/ Windows Phone 8.1, no Lumia Cyan.)
The biggest problem for me is that I live in Canada where we do not get the Lumia 1520. I had to bring in the international unlocked version, which I love because of the built-in 32 Gb of storage and built-in wireless charging, but it also means that I cannot get warranty work on it here. I even looked into it in the US too and Nokia there also says that they cannot perform warranty work on it because it is not a US AT&T model. Frustrating. I am not in a position to afford to replace this phone after only 6 months. I might be going back to my Lumia 920 in the meantime, which has a broken vibrate motor but at least doesn't bounce all over the place.
I've fiddled with screen sensitivity and my double-tap to wake is currently disabled, so it's not that. I'm just stumped. I might try nuking it a couple times and see what I can find. Also, squeezing it doesn't really seem to fix it, nor tapping it on my arm or leg. It might be doing it more in landscape mode, as the worst was this morning, when I had to RDP into a work server to resolve something. Again, might just be coincidence.
Curious to see how many others have this issue and if anyone might have any other tricks up their sleeves!