Linoleum is one of the more flexible types of flooring material you can use in your home, but it still has to be installed on a rated type of substrate for the proper finish result and warranty coverage. Just like ceramic tile, linoleum needs to be installed on a flat surface that will bond with the adhesive used in installation, such as plywood or cement, but not carpet or carpet padding.
Linoleum flooring material is almost exactly like ceramic tile in its installation protocol. A flat, clean surface is required for the tiles or sheets to be installed properly, and they are installed against each other within an overall pattern. Linoleum cannot bend more than just slightly, otherwise the product will snap in half, plus the tiles butt up against each other so lippage is a concern. A flat surface is required.
While the surface does need to be flat and capable of bonding with whatever adhesive you have chosen to use with your linoleum, it also needs to be a type of underlayment that is approved for use with linoleum. As a general rule, linoleum doesn't have the weight or stability requirements of ceramic tile, so concrete board underlayment isn't required. Instead, plywood, concrete or any other type of flat surface will do.
Padding for carpet is in no way flat enough to provide an adequate surface for gluing down linoleum. It is also far too flexible to accommodate any type of hard or semihard flooring surface on top of it, because the cushioned aspects of the padding would lead to a person stepping on a piece of linoleum tile that sinks into a cushioned spot beneath the weight of an individual.
The Bare Basics
For best results when dealing with a remodeling project, you need to strip the carpet, the padding and everything else all the way back down to the original rough flooring for the home, which is probably particle board or plywood. Once you have finished, you can remove all the staples and nails and ensure the floor is flat enough to begin the installation.