It doesn't have to literally be a desert, just somewhere dry. In fact, somewhere uncharacteristically dry (i.e. suffering from a drought) or where the lack of water is key would be ideal.
There must be lots of films which show rabbits in the Australian outback, but obviously they tend not to be very central to the plot.
I was wondering about Bugs Bunny. Some sources seem to suggest that he is even from the desert, but I can't identify a film where this is important.