Vinyl covered cabinets
You'll want all of the adhesive off before you paint. If you want a non-chemical alternative, you can scrape off the majority a lot easier if you heat it up with a heat gun or a hair dryer.  This is basically what your sander is doing through friction, except the melted adhesive is sticking to the sandpaper and smearing instead of coming off.

After you get the bulk of it off, the remaining residue can be stripped off with an adhesive eraser wheel in an electric drill (do this outside, it will make a mess). This is essentially the same thing as the belt cleaner @paperstreet recommends - just powered by a drill instead of an elbow. Auto supply stores would be a good place to look for one.

Finish up by sanding with 120 or 150 grit sandpaper followed by 220 grit. 

BTW, I hope that you told Sherwin Williams you were painting MDF and that they didn't give you a water based primer - these can raise the fibers in MDF and make it practically impossible to sand out.