Chipping dead wood makes a mess. Lots of broken bits flying around, lots of dust depending on just how dry the wood is. Dead wood can also jam up a chipper wheel in a hurry if you feed too fast. Live wood makes nice clean chips that flow, dead wood just shatters and the resulting larger bits can wedge the wheel tight if you aren't careful. Don't count on dead wood for cover mulch either. Even if you do manage to get some decently consistently sized chips, chances are good that they could contain insects, fungus or other things that live in dead wood that you don't want to bring near your house or garden.
Dead trees don't always cooperate when you drop them. Cutting a live tree you can get a nice flexible hinge to direct the fall, but dead dry wood can just snap off as soon as it starts moving and fall in just about any direction. Sometimes they just won 't fall at all and need some input from you to get them moving. Problem is pushing or pulling a long dead tree can break it off and send it in any random direction. When it hits the ground, even a dead tree that looks solid can disintegrate into a million bits, especially if it still has a bunch of small branches. Lots of clean up by hand will result, and possibly not much left worth chipping either.
Just curious, have you compared the cost of having a few yards of good clean garden mulch delivered with the cost of doing this yourself? I don't know what sort of experience you have with this sort of stuff, but one guy with a saw and a 6" chipper isn't going to be able to cut and chip enough wood to get anywhere near 10 yards in a day. There is a lot more involved than throwing branches through the chipper. You'd need to have a bunch of stuff cut and trimmed to fit the chipper before hand, properly stacked so its easy to work the pile. It takes a considerable amount of branches to make 10 yards of chips if you figure something like a 20:1 reduction rate.