There’s an incredibly simple effect that can be done to convert animated GIFs into 3D.
All you need to do is put two solid-coloured vertical lines onto the image (as shown by the two white bars left). As objects move in front of the white lines, it appears as if they’re coming out of the screen, extending beyond the image, just like 3D.
A range of other 3D GIFs can be found over at My Modern Metropolis. It’s rather funky. Onwards!