EDIT: This bug seems to have resurfaced in 15w42a, under slightly different conditions.

To reproduce, run the command

/summon ArmorStand ~ ~1 ~ {ArmorItems:[{},{},{},{id:skull,Damage:3,Count:1,tag:{SkullOwner:{Id:"Test"}}}]}

in a command block. ArmorStand may be replaced with any entity capable of wearing armor, and {Id:"Test"} with any NBT that produces invalid texture information.

The bug seems to behave exactly the same as before, except that hitting the affected entity now has a chance of crashing the game immediately. The red screen and pixelated main menu effect now clears up if an unaffected mob is damaged, but it is unclear whether this also happened in prior releases.

When the game attempts to render any entity wearing a player skull with the data tag SkullOwner:"", it bugs out and begins to render entities, tile entities, and in extreme cases transparent blocks with incorrect locations and rotations.

To reproduce, run the command

/summon ArmorStand ~ ~1 ~ {Equipment:[{},{},{},{},{id:skull,Damage:3,tag:{SkullOwner:""}}]}

in a command block. ArmorStand may be replaced with any entity capable of wearing armor.

FULL DETAILS:
First of all, if the player enters the same render chunk as the affected entity while in third-person mode, they will appear in the same place as the entity, upside down. When the player is not in the rendering chunk, the skull appears to latch onto an entity or entity part and renders that at its location. For example, I observed a bow with no skeleton and a hopper minecart without its hopper. I noticed that the minecart would not render if its original location was not in view. The skull would latch onto entities even if they were not in the same rendering chunk.

The skull also causes entities and tile entities (such as chests) in the vicinity to appear upside down and on the other side of you, and move when you move. It appears that the entities are being rotated around you. Which entities were affected varies based on where you are standing and where you are looking. Everything goes back to normal if you don't look at the spot where the cursed skull is. Interestingly, when the skull latched onto the hopper minecart, the hopper part continued floating around in the sky.

Summoning a cursed entity, killing it, and summoning a new one without restarting the game still causes entities to become flipped, but does not cause anything to appear specifically at the skull's location, except for a split second when exiting the render chunk.

Cursed skulls exhibit no strange properties when held in a player's hand nor when placed on the ground. When worn by a player the player's inventory starts rendering items in the wrong locations, and when the player is in third-person mode so that the skull is rendered, all normal glitches still occur. Additionally, while a cursed skull is being rendered on any entity, parts of the GUI may disappear when the player's hand is empty. The other parts of the glitch also seem to vary based on whether the player is holding an item and whether the player is in third-person mode.

Summoning two cursed entities causes the rendering bugs to become much worse. All entities, including both cursed entities, are offset and rotated. Transparent blocks and the block outline cursor are also affected. Adding three or more has no further effect, beyond a possible increase in the amount of offset.

Also, all the effects listed above still happen when the nameless head is on a Zombie, but the affected entities are rotated and offset based on the Zombie's movement as well as the player's. When the Zombie dies, the affected entities all rotate with it as it falls down, then return to normal when it disappears. Hitting the Zombie causes all worlds to be tinted red until exiting the game and rebooting it, and also makes the background of the main menu low-resolution.

After exiting the game, the console logs show this message repeated over and over:

[00:07:45] [Client thread/ERROR]: ########## GL ERROR ##########
[00:07:45] [Client thread/ERROR]: @ Post render
[00:07:45] [Client thread/ERROR]: 1283: Stack overflow
[00:07:45] [Client thread/ERROR]: Couldn't render entity
java.lang.IllegalArgumentException: Name and ID cannot both be blank
    at com.mojang.authlib.GameProfile.<init>(GameProfile.java:25) ~[authlib-1.5.17.jar:?]
    at csh.a(SourceFile:81) ~[1.8.jar:?]
    at cqv.a(SourceFile:411) ~[1.8.jar:?]
    at cqv.a(SourceFile:170) [1.8.jar:?]
    at cqz.a(SourceFile:40) [1.8.jar:?]
    at crz.a(SourceFile:53) [1.8.jar:?]
    at crz.a(SourceFile:13) [1.8.jar:?]
    at cpt.a(SourceFile:294) [1.8.jar:?]
    at cpt.a(SourceFile:259) [1.8.jar:?]
    at cpt.a(SourceFile:226) [1.8.jar:?]
    at ckn.a(SourceFile:583) [1.8.jar:?]
    at cji.a(SourceFile:1225) [1.8.jar:?]
    at cji.a(SourceFile:1130) [1.8.jar:?]
    at cji.b(SourceFile:983) [1.8.jar:?]
    at bsu.as(SourceFile:903) [1.8.jar:?]
    at bsu.a(SourceFile:314) [1.8.jar:?]
    at net.minecraft.client.main.Main.main(SourceFile:120) [1.8.jar:?]