I've read flint and obsidian. But surely those are too brittle to shave with. Obsidian with its sharpness would probably tear up your face.
Also some of the earliest razors were found in Egypt. Does anybody know if razors were invented independently anywhere else (the Nordic Bronze Age had some I believe). Also where in Egypt are these razors from?
Here's a response to a similar question I answered last year, focusing on Native North America and the use of stone tools for cutting hair and shaving. Note the reference from Fray Juan de Torquemada about the Native Mexicans and the early Spanish colonizers using obsidian blades for shaving.
tldr: a finely-knapped stone tool works fine for shaving and isn't any more dangerous than a metal razor, but they do break and dull more easily than metal; other ancient hair removal methods included plucking and abrasive or adhesive depilatories ("waxing").