Dick, this varies some with the linoleum. If it's a relatively thin, flat surfaced linoleum you're probably okay, but to be on the safe side using a thinner backer, I'd pull it. Many linoleums are relatively easy to remove, and separate similarly to strippable wallpaper; leaving the paper backing. A good way to start is by cutting the vinyl into strips about 2' wide with a utility knife, get an edge started, and pull up the strip. If there were any areas which didn't adhere well, the backing may come up, too, and these areas should be skimmed with floor patch prior to installing your 1/4" backer.
When it comes to installing the tile, use a full-flex mortar.