This paper establishes sharp nonexistence criteria for nonnegative solutions to a class of quasilinear elliptic inequalities and divergence-type systems in the subelliptic framework of the Heisenberg group $H^n$. By developing an optimized test function methodology adapted to the stratified Lie group structure, nonexistence is established through a contradiction argument based on maximum principle-type inequalities. The analysis contributes new insights into the role of sub-Riemannian geometry in constraining the solution behavior for degenerate elliptic operators.