Although the most common answer to questions like that is "relative to what?", there are possible side-effects. If I recall correctly, the earth's velocity relative to the cosmic background radiation is on the order of a mere 600km/s. If we were to be travelling arbitrarily close to the speed of light as compared to our current motion, part of this background radiation could be blue-shifted to such an extent as to be harmful to us.
The answer remains: it depends, relative to what you will be travelling at the speed of light to. But if it's relative to now (as in "current state"), there are plenty of things in the universe that you wouldn't want to interact with.