Only if you can define a meaningful semantics for it.  Lists aren't a monad because of liftIO, but because the monad pattern means something (backtracking/multiple evaluation).  What does the monad pattern mean for your heap?

I'm not saying there *isn't* one; I'm saying that you must *have* one in order to make a monad.  Monads are not simply some sneaky way to slipstream I/O into expressions; IO is just one possible monad, and many programs make heavier use of other monads such as state and reader.  And in particular, most applications of the list monad don't involve any I/O.  (Almost necessarily, since the standard ListT is actually broken but can't be fixed for backward compatibility reasons.)