It's actually a myth that teeth whitening will do serious damage to the enamel in your teeth. It won't do anymore damage than some of your favorite foods already do.
However, one often mentioned side effect of teeth whitening is increased tooth and gum sensitivity. This usually goes away once you stop whitening your teeth though.
Another side effect is acid burn. If you're using a poorly designed dental tray or whitening strip that isn't properly fitting your teeth, its possible for some of the hydrogen peroxide to touch your gums. If this happens your gums will turn white temporarily until it heals.