jmc

Since the GCD of $m$ and $n$ is 8, $m=8x$ and $n=8y$ for some integers $x$ and $y$. Note that minimizing $m+n=8x+8y=8(x+y)$ is equivalent to minimizing $x+y$.

The LCM of $m$ and $n$ is $112=2^4\cdot 7=8\cdot 2\cdot 7$, so one of $x$ and $y$ is divisible by 2 and one is divisible by 7. Then we can minimize $x+y$ by setting $x$ and $y$ to be 2 and 7 in some order. Therefore, the least possible value of $m+n$ is $8(2+7)=\boxed{72}$.