It depends on which lysol product you are referring to. If you mean the spray then yes it can be dangerous - very dangerous if used near the birds - as is any aerosol. But if you mean just the disinfectant I often use that to clean Auggie's cage, though he's not in it at the time, it is well rinsed and dried before he goes back to it.
As far as I know lysol is safe with exceptions: stay away from aerosols and don't use the lysol by the bird(s).
Why not use the disinfectant wipes to clean his cage? You can wipe it down with a damp paper towel afterward to make sure you're not leaving any disinfectant behind, and then just let it dry.