Of all editions of D&D however, 5e is the least affected by lack of magical healing. So there’s that...You can run most any flavor of D&D sans magic. It's going to change around the dynamics of the game. Anyone running such a game needs to carefully consider which monsters and abilities are used. Healing will be hugely reduced, so the game would most likely need to have most of the "resource depletion" encounters removed, and focus on a smaller number of more focused, storyline important encounters. Probably need to change around advancement as a result.
Because there won't be any "magic items", treasure will become far less important. This will also mess with various "CR" systems for balancing encounters.
I think it can be done, but to be a successful game it's going to take some thinking on the part of the GM.
Personally, for such a game I'd skip any flavor of D&D and opt for something like Mythras.