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.
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Fixed in 1.8.1-pre1

This is NOT fixed in pre1 or pre2, nor is it a dupe of 1279, please reopen.

Please attach a zip of a world showing that issue.

OK, I've uploaded a pic showing that this still affects pre2, and attached a world. Please reopen.

Note: All of my worlds that feature this are far too large to upload here, so I had to do some more digging. After reading MC-1279 in more detail, it mentions that the beacon slow rotate issue is due to the world's time. I realized that in my worlds there was a correlation between world age and how garbled the texture got. After some more investigation and testing with NBTExplorer, I confirmed that high values on the 'time' parameter caused the problem. For the purposes of this bug, I created a superflat world and manually set 'time' to 19999999 (and took a screenshot).
As far as dupes go, MC-1279 is about how beacon beams' rotation animation gets progressively slower and more choppy with world age. I have never seen slow or choppy beams in any of my worlds, instead the beam texture gets progressively more distorted. Whatever fix was put in for MC-1279 didn't resolve my issue. While these two bugs are definitely closely related, they're not exactly the same thing, however if you want to reopen MC-1279 and then roll this bug into it, I'd be fine with that.

quartz, MC-1279 is actually fixed for a future version. 😛
Otherwise, cannot confirm with the attached world. Please see attached screenshot.

OK, well, at the time Kumasasa responded with "Fixed in 1.8.1-pre1" MC-1279 was listed as being fixed in pre1, not future-1.8.2, so I'm not sure what's going on there.
As far as confirmation goes, I forgot to attach a resource pack with the cut up texture I was using (the problem is still there with the vanilla texture, but hard to see). I'll attach that now and maybe it'll be more obvious (it's worth mentioning that the flickering garbled-ness is much more apparent when actually playing the game, you can't really capture how bad it looks in a screenshot).
Also, being a texture/rendering issue, there's a good chance it might be hardware specific.

The attached resource pack has two textures in it. "beacon_beam.png" is the rainbow one that doesn't show up at all, whereas "beacon_beam_slices.png" is the white one.

Tested and confirmed with beacon_beam.png, but cannot confirm with beacon_beam_slices.png.

More playing around and testing stuff. I've updated the description with a second edit. I don't know what the right procedure is in this case.

Sonic: what version were you using when you [partially] confirmed?

1.8.1-pre2, quartz.

Is this still an issue in the current Minecraft Snapshot 15w47c or later? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.

Ticket resolved as incomplete, because no answer in a reasonable amount of time (1+ year), if it still happens, please update the ticket.