The influence of long-range interactions on the stability of stationary solutions of triangular lattices described by the continuum-discrete nonlinear Schrödinger equation is analyzed. By virtue of the linear stability analysis and a variational approach we demonstrate that both soliten array and continuouswave solutions are modulationally unstable. Analytical expressions for instability thresholds and growth rate spectra are presented and compared with the corresponding results in the approximation of a nearest neighbor interaction.