Drying- I let these burls dry inside my shop for over a year. They had been drying covered outside for several years. If you open up the wood while the moisture content is too high you are asking for it to split on you. This can be lessened to some extent by spraying it with a coat of poly when you are done each night but it can get irritating sanding it off in the morning. You could also keep the wood in plastic overnight.
Pitch- I only mentioned it briefly but as the burl is scar tissue it tends to be full of pitch. The surface layers will dry and harden over long periods of time (read months) but as you sand and cut you expose soft layers soaked into the wood. Now those too will harden over long periods of time but who wants to wait? If you spray poly or lacquer over it the solvents will soften the resin and lift it, marking the finish. I attacked it two ways...
Turpentine: Soak and scrub, soak and scrub. This will lift the pitch to the surface then you can scrub it off with a rag. This will only take up surface deposits but still that might be enough. On my table this is what I did.
Bake it: Find a local kiln and ask that they toss your dry burl in to set the resin. They do that type of thing all the time and the price will be cheap. This will harden the resin inside the entire chunk. It will still soften if you create a lot of heat sanding so go slow and keep it cool.
Seal it: Spray your bad spots with shellac to seal them in. I would only do this if the spot was minor. Once you have a layer or two of shellac protection built up finish normally. This will give the wood an extra yellow/orange look.
PLEASE NOTE! ON A HOT DAY THE RESIN IN THE WOOD COULD STILL SOFTEN AND WELL UP THROUGH THE FINISH CREATING STICKY CRACKS. (it can even do this though poured epoxy) THE ONLY THING THAT WILL STOP THIS IS TO HAVE THE PIECE KILN DRIED TO SET THE RESINS.