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An excitonic insulator (EI) phase is a consequence of collective many-body effects where an optical band gap is formed by the condensation of electron-hole pairs or excitons. We report pressure-dependent optical pump optical probe spectroscopy of EI Ta$_2$NiSe$_5$ in an on-site in situ geometry. The fast relaxation process depicts the transition across P$_{C_1}$ $\sim$1 GPa from EI phase to a semiconductor and P$_{C_2}$ $\sim$3 GPa from a semiconductor to a semimetallic phase. The instability of the EI phase beyond P$_{C_1}$ is captured by the Rothwarf-Taylor model by incorporating the decrease of the bandgap under pressure. The pressure coefficient of the bandgap decreases, 65 meV/GPa closely agrees with the first principle calculations.