$\because |m|=3$，$|n|=5$，

$\therefore m=\pm 3$，$n=\pm 5$，

$\because m-n \gt 0$，

$\therefore m=\pm 3$，$n=-5$，

$\therefore m+n=\pm 3-5$，

$\therefore m+n=-2$或$m+n=-8$.

故选$C$.