Why is the impulse response function of this system 0? Suppose I have an system $ y(t) = t^{2}x(t)$. 

The impulse response of this system would be: $h(t) = t^{2} \delta(t)$. 

Since $\delta(t) = 0$ for $t \neq 0 , h(t) = 0$ for $t \neq 0$.
And at $t=0, h(t) = 0$. 

But since $y(t) = \int_{-\infty}^{\infty}x(r)h(t-r)dr $, wouldn't $y(t) = 0$?

What am I missing here?
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.