jmc

Since $2000,2001,\dots ,2006$ are $7$ consecutive integers, they include exactly one integer from each residue class $(\mathrm{mod}7)$. Therefore, their sum is congruent $(\mathrm{mod}7)$ to $0+1+2+3+4+5+6=21$. The remainder of this sum $(\mathrm{mod}7)$ is $\boxed{0}$.