Sauron doesn't turn invisible because he is above the material world. He is of the beings that helped make it.
You have Eru Ilúvatar who if you were to compare to Christianity, would be God. You have the Ainur who are equivalent to angels for all intents and purposes. The Ainur split into two groups, the Valar (The greater of the two) and the Maia (The lesser spirits). I say they can be called angels, but the Valar might even parallel the Greek gods.
Sauron is a maia, just like Gandalf, Saruman, the Balrog, etc. The Ring was created by him as a means to control the other rings he offered to the Elves, Men, and Dwarves. He had complete control over the One Ring because he was its creator and master. I imagine any Maia would have had the abilities or know-how to demonstrate some form of control with the ring, but the theory never gets tested, as Gandalf is too afraid to use it. (We can only speculate that he was only too weak to handle it in his human form, but maybe even Olorin (Gandalf as a Maia, not a man) could have been corrupted) We can also only speculate that Tom Bombadil, the only potential Ainur to have donned the ring besides Sauron, was even a Maia to begin with.
But Tom Bombadil did clue is into one thing, and that is that not ALL beings were drawn by its corruption, thus leading me personally to believe he was either Eru himself, or that if Gandalf had been in his true form, he would not have had problems with the Ring. As a manifestation of man, perhaps he was more prone to be corrupted. (A trait we see mirrored in the fact that the nazgul were the only ones to fall corrupt) (Edit: Oh and also in Saruman's betrayal)
Of course my TL;DR answer would be that once you've turned your back on your creator, and your big brother Morgoth is gone, and dissolved your soul into a material possession, what is left to turn invisible :D