Does thermal paper darken if left in the sun? If not, I don't see how a projected image will be hot enough to burn an image. Even if it warms up the paper, the process has to happen fast enough that the heat isn't conducted to the surrounded area first. Maybe you could make a solargraph with thermal paper, where the sun burns a path over many days.

Typical photographic paper (developed in the darkroom) or direct positive paper is better suited to what you're suggesting.