If the universe was infinite, would the sky not be completely illuminated with light?

No. This was actually an important argument thought up several hundred years ago to demonstrate that the Universe is necessarily finite in either age or size (or both). We now know that the Universe is finite in age (it is about 13.7 billion years old), and moreover is constantly expanding in such a way that, even when its age becomes infinite, the sky will still be black. We are not yet sure whether it is finite in size, although if it is, its minimum size is still very, very big.