Tornheim-like series, harmonic numbers and zeta values

Explicit evaluations of the Tornheim-like double series in the form $$\sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s \right)},\ s\in \mathbb{N\cup } \left\{ 0 \right\}$$ and their extensions are given. Furthermore, series of the type $$\sum_{m=1}^\infty \frac{2H_{2m+1}-H_{m}}{2m\left( 2m+1 \right)}$$ and some other Tornheim-like multiple series are evaluated in terms of the zeta values.