We present magnetization measurements on polycrystalline $beta$-Li$_2$IrO$_3$ under hydrostatic pressures up to 3 GPa and construct the temperature-pressure phase diagram of this material. The magnetically ordered phase with $T_{rm{N}}simeq 38$ K breaks down upon a pressure-induced first-order phase transition at $p_{rm{c}}$ $approx$ 1.4 GPa and gives way to a high-pressure phase, where a step-like feature in the magnetic susceptibility signals a structural dimerization with a loss of Ir$^{4+}$ magnetic moments. Nevertheless, magnetism manifests itself also above $p_{rm{c}}$ via the Curie-like susceptibility upturn with the effective moment of 0.7 $mu_B$. We suggest that a partially dimerized phase with a mixture of the magnetic and non-magnetic Ir$^{4+}$ sites develops above $p_{rm{c}}$. This phase is thermodynamically stable between 1.7 and 2.7 GPa according to our ab initio calculations. It confines the magnetic Ir$^{4+}$ sites to weakly coupled tetramers with the singlet ground state and no long-range magnetic order. Our results rule out the formation of a pressure-induced spin-liquid phase in $beta$-Li$_2$IrO$_3$ and reveal peculiarities of the magnetism collapse transition in a Kitaev material.
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