Note before you comment, I do not have the animated water setting off.

When the player is on fire and is in first person view the fire shows up in front of the screen to show the player is burning. (Normal) But the fire does not animate if there is no animated textures in view such as water, fire, lava, ect..

To reproduce

Catch fire.
See the fire animting. Move away from animated textures such as water, nether portals, lava, fire, ect...
The fire animation on the screen freezes.

I Would expect it not to freeze.
Video shows a reproduction of the bug.