Long run trade off between inflation and output
Note that if $\pi_t = E_t\pi_{t+1}$ (current and expected future inflation are equal) then

$$\pi_t = E_t\pi_{t+1} + h(y_t-y_t^*) \implies 0 = h(y_t - y_t^*) \implies y_t = y_t^*$$

So as the output gap tends to zero in the long run, inflation will perfectly determine next period's inflation, so monetary policy that affects inflation in the short run cannot affect output in the long run. Thus there is no tradeoff between inflation and output in the long run.