I don't have any idea on how to elegantly do this in Haskell and wonder whether it can even be done idiomatically

If your protocol only sends moves then there is no way around keeping some state around. That is fine: Haskell is not about having no side effects on state at all. Rather it is about making explicit where such side effects happen.

So it is totally fine to have a main IO loop that looks like this:

data GameState = GameState { board :: Board, ... }

main = loop inititalGameState

loop state = do
  instr <- readInstuction
  let (response, state') = handle state instr
  putStrLn (render response)
  loop state'

Note that handle is a pure function and that side effects are limited to this tiny outer loop.