We experimentally and theoretically investigate the electrically-tunable third-harmonic generation in the mid-infrared in a free electron liquid that can be described by a hydrodynamic model. Grating-gate field effect devices fabricated on a heavily n-type doped InGaAs/InP heterostructure have been pumped with ultrashort mid-IR pulses. A dependence of the third-harmonic generation efficiency on gate voltage bias has been observed. Our findings pave the way for future exploitation of tunable free-electron nonlinearities in semiconductor-based photonic integrated circuits incorporating gated heavily-doped semiconductor plasmonic waveguide sections.