I'd say discuss this with your veterinarian. I'd likely ask doc to thoroughly examine the teeth to determine if they remaining are healthy or not. If his teeth are suspect, then I'd discuss how hard it would be to remove them all. Sometimes their teeth become loose and infection sets in. The cause could simply be genetic or it could be due to bad diet during development, its hard to determine.
I have had a couple of toothless hedgehogs in the past and they lived quite well on a soft food only diet. The biggest problem with soft food only diets sometimes is getting a picky eater to want to switch.