One application is to computer graphics. When simulating the effect of lighting a translucent material (see my avatar!) you often need to integrate over all possible paths from the light source to the camera via the material. This is similar to the Feynman integral in quantum mechanics, but note that this is an integral in the domain of classical geometric optics, not quantum field theory.
I believe it was Jerry Tessendorf who pioneered this approach in the graphics world. You may have watched movies with effects rendered using techniques derived from Tessendorf's!
I should add that this is a particular case of what Steve Huntsman describes in his answer.
