jmc

Let $a,$ $b,$ $c$ be the roots of $x^3-49x^2+623x-2015.$ Then by Vieta's formulas, $a+b+c=49.$

The roots of $g(x)=f(x+5)$ are $a-5,$ $b-5,$ and $c-5,$ and their sum is $a+b+c-15=49-15=\boxed{34}.$