Good question! I'm afraid I don't know if any diseases are directly transmissable between birds and rabbits, but I guess it might increase the risk of birds bringing in diseases such as VHD on their feet (amongst the many other ways it could be brought in e.g. on hay, clothes, wind, dogs etc.). As with everything, I think it's a case of minimising risk!
For example, you could confine the feeders to one small area so it's easier to disinfect (using a safe cleaner such as Ark-Klens) or even exclude your rabbits with a bit of fencing. Or maybe put the feeder above something that will catch droppings e.g. hang the feeder from a hanging basket bracket on a fence, with a 'shelf' below).
Maybe use mesh feeders so that whole seeds/nuts aren't dropped on the ground that the bunnies could eat (although in my experience the scraps get hoovered up pretty quickly by ground-feeding birds!). Choose the seed mix carefully, lots of standard mixes have dried corn etc. (and check none of the ingredients are toxic to bunnies)
We have several bird feeders in our garden, but have put them in trees in the corner which we block off because the leaves probably aren't bunny safe. I think the birds appreciate an extra food source in this weather!