This gif strongly suggests that at least cats do:

This page will show you a larger version of the “wheel” illusion that the cat is inspecting (and attacking). I think it’s very clever to present such images to animals who can show us directly (as hunting cats can) whether their senses are fooled. This moggie was clearly duped!
That’s not all that surprising, of course, as we naturally think that other mammals’ eyes work the same way as ours. But this shows more than that—it shows that other mammals’ brains work the same way as ours (at least insofar as the illusion works), for what is taken in by the eye is interpreted by the brain. This cat sees a nonmoving image and, like us, interprets it as being in motion.
Optical illusions are a byproduct of the evolved interaction between vision and its neural interpretation, and many such illusions—like the “checker-shadow” illusion, perhaps the best of all optical illusions— reflect our evolved ability to compensate for natural phenomena. Greg Mayer has written about how countershading in animals takes advantage of predator’s evolved tendency to be fooled about the hue of an object. Likewise the checker-shadow illusion fools us because our brains evolved to imagine things in shadow to be lighter than they really are.
Of course many of the selective pressures that molded our own interpretation of what we see must be similar in animals. Predators and humans are both fooled by countershading, and I bet a mammalian predator’s brain would also be fooled by the checkershadow illusion. (Actually, that could be tested, at least in birds, by training them to peck at squares of a certain hue and then giving them the illusion.) Or perhaps we’ve simply inherited the eye-brain connection that is fooled by illusions from our ancestors in which that visual interpretation was critical for survival, and we’re the victims of “evolutionary inertia.” Regardless, the cat above shows that we’re not the only species fooled by these two-dimensional tricks.
h/t: John S.