If H(m) = 0 for some m, how can a DSA signature be forged?
The signature equation is $$r = (g^{H(m) s^{-1}} y^{r s^{-1}} \bmod p) \bmod q,$$ where $g$ is the standard generator of an order-$q$ subgroup of $\mathbb Z/p\mathbb Z$, $y \in \mathbb Z/p\mathbb Z$ is a public key, $m$ is a message, and the signature is $(r, s)$ for scalars $r, s \in \mathbb Z/n\mathbb Z$.

If $H(m) = 0$, this reduces to $r = (y^{r s^{-1}} \bmod p) \bmod q$.  If you pick $r = s = y \bmod q$, then $r s^{-1} \equiv 1 \pmod q$, so $y^{r s^{-1}} \bmod p = y$ and so $(y^{r s^{-1}} \bmod p) \bmod q = y \bmod q = r$ as desired.