Simplify[Conjugate[Exp[I h t]], h t ∈ Reals]
    (* E^(-I h t) *)

`Simplify` can handle this, if the assumption that the exponent is purely imaginary holds. Note that this assumes that the product is real: if each variable is real, you may wish to state that instead:

    Simplify[Conjugate[Exp[I h t]], {h ∈ Reals, t ∈ Reals}]

If `Simplify` takes too long it's also possible that you could consider an equivalent replacement rule:

    Conjugate[Exp[I h t]] /. Conjugate[h t]->h t

However, you'll have to be a bit more careful with this, since the inside term can vary quite a bit based on assumptions, variable ordering, etc.

If you are not certain that $Ht$ is real, then it's not actually necessarily true that $e^{iHt}=e^{-iHt}$.

And if you are *absolutely*, positively, and perfectly certain that the conjugate of an exponential containing `I` is purely imaginary, you can do this:

    Unprotect[Conjugate];
    Conjugate[Exp[x_ /; Not[FreeQ[x, I]]]] := 
        Evaluate[Exp[Conjugate[x]]] /. Conjugate[y_] -> y;

That will cause issues in a variety of places, so double check the use case. You will likely need to refine the qualifier (`Not[FreeQ[x,I]]` is almost definitely overzealous).