Beacon beams render as garbled flickering static, distorted, or totally missing to varying degrees, depending on the world. Some worlds the beams render fine, some are a total mess, and some are in between. There appears to be no rhyme or reason as to which worlds have problems and to what degree they have them. However, the issue is totally reproducible per-world; any world that shows a problem maintains it to the same degree always, regardless of the number of times I quit and relaunch. Destroying and re-placing the beacons or any surrounding blocks has no effect.

The problem is hard to notice with the vanilla textures, so I've attached six examples: two pictures each of a custom texture on three different worlds. World "B" appears fine, world "A" has medium distortion, and world "F" has very bad distortion as well as the rainbow variant not showing up at all. In each case, I've superimposed the custom texture in use over the right hand side of the image so things are easier to understand.

NOTE: This is NOT a dupe of MC-68247, as this bug has nothing to do with viewing angle. In the case of "F-rainbow", all beams of all beacons in the world are missing all the time, no matter where you are or where you're looking.

EDIT
It turns out there is a direct correlation between the age of the world (NBT 'time' parameter') and how bad the texture looks. I can make the problem happen or disappear by manually editing that with a program like NBTExplorer. I'm not sure if this qualifies as being a dupe of MC-1279 though, as that bug is mainly about the beam animation being slow and choppy, but I've never seen that.

EDIT2:
I can get slow and choppy beam rotation if I edit the 'time' parameter to a very high value, and then look directly at the beam and stay completely stationary for several seconds (the animation seems ok when I move around). It looks like on my system that the texture gets garbled long before the animation gets choppy, and none of my natural worlds were quite old enough to exhibit that. This bug and MC-1279 are a lot more closely related than I initially thought. I'm not sure what the deal is with MC-1279 being closed though (was it really initially fixed for pre1 or did someone put that in by mistake and it's not actually fixed until 1.8.2?). Sonic partially confirmed seeing it too (on only one of the textures), but I'm not sure what that means for this bug. I honestly don't know if this should be left open, or MC-1279 should be reopened, or what.