I think Joe is talking about an oscillating multi-tool and in this instance he is using it more like a power scraper. Mine is a Fein and one of the best tools I have ever owned. Pricey but Rockwell, Craftsman, Dremel (I think) and even Harbor Freight make similar units. Check this site for reviews. Some have been happy with the brands they got but I forget which topped the list of also rans.
You do want to get most all the adhesive off or you may have adhesion problems. Most certainly you need a level surface with no ridges to lay tile flat. Without knowing what is on there it is hard to recommend a solvent to get the final residue off. Laminates were usually applied with contact cement on both surfaces but you said this is the backsplash on plaster?