Why is the impulse response function of this system 0?
This system 
$$ y(t) = t^2 x(t) $$

is not LTI and therefore does not have an impulse response of the form $h(t) = \mathcal{T}\{\delta(t)\}$.

So your statement $h(t) = t^2 \delta(t)$ is not correct... Hope this solves your confusion.