By multiplying out the factor $H_n^3$, it is not too hard to see that your sum can be written as a rational linear combinations of special values of weight $4$ multiple polylogarithms. The Maple package HyperInt (by Erik Panzer) can perform simplifications with multiple polylogarithms. According to this software, $$ \sum_{n=1}^\infty \frac{H_n^3}{(n+1)2^n}=\frac{\log(2)^4}{12}-\frac{\pi^4}{144}+\frac{\log(2)^2\pi^2}{6}+\log(2)\zeta(3). $$

By multiplying out the factor $H_n^3$, it is not too hard to see that your sum can be written as a rational linear combinations of special values of weight $4$ multiple polylogarithms. The Maple package HyperInt (by Erik Panzer) can perform simplifications with multiple polylogarithms. According to this software, $$ \sum_{n=1}^\infty \frac{H_n^3}{(n+1)2^n}=\frac{\log(2)^4}{12}-\frac{\pi^4}{144}+\frac{\log(2)^2\pi^2}{6}+\log(2)\zeta(3). $$ The algorithm the software uses is described in Panzer's paper "Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals" (there is also a version of the paper on arXiv).