Can I replace a 5600uf 6.3v capacitor with a 4700uf 35v in a DirecTV tuner?

From the voltage rating, I'd say it's OK. From the value rating, there's usually a reason a particular value's chosen, but it shouldn't hurt to try. SOMETIMES,a nd I really do mean, sometimes, you can get away with it. It just depends on the application it's being used.

Because of the inherent wide tolerances(20% or more) of electrolytic capacitors, this will probably work just fine.