Could a marshmallow traveling at the speed of light knock the Earth out of orbit?

The answer is, no, such an impact would not have any measurable effect on the Earth's orbit.

If a typical marshmallow weighs 20 grams, then from E = mc^2 you get an energy of 20 x (3x10^10) x (3 x 10^10) = 2 x 10^22 ergs.

I believe this is about equal to the yield of 5 - 10 hydrogen fusion bombs.