No. Take a triangle $v_0v_1v_2$ on the plane, $H'$ is its side $v_0v_1$, $H$ is almost another side $v_0v_2$. Then $\sum_{i=0}^2 {\rm dist}\, (v_i,H')$ is just a length of altitude from $v_2$, $\sum_{i=0}^2 {\rm dist}\, (v_i,H)$ is almost the length of altitude from $v_1$, which may be less than that from $v_2$.