Heating effects in Floquet-engineered systems are detrimental to the control of physical properties. In this paper, we show that the heating of periodically driven strongly correlated systems can be suppressed by multicolor driving, i.e., by applying auxiliary excitations which interfere with the absorption processes from the main drive. We focus on the Mott insulating single-band Hubbard model and study the effects of multicolor driving with nonequilibrium dynamical mean-field theory. The main excitation is a periodic electric field with frequency Ω smaller than the Mott gap, while for the auxiliary excitations, we consider additional electric fields and/or hopping modulations with a higher harmonic of Ω. To suppress the three-photon absorption of the main excitation, which is a parity-odd process, we consider auxiliary electric-field excitations and a combination of electric-field excitations and hopping modulations. On the other hand, to suppress the two-photon absorption, which is a parity-even process, we consider hopping modulations. The conditions for an efficient suppression of heating are well captured by the Floquet effective Hamiltonian derived with the high-frequency expansion in a rotating frame. As an application, we focus on the exchange couplings of the spins (pseudospins) in the repulsive (attractive) model and demonstrate that the suppression of heating allows us to realize and clearly observe a significant Floquet-induced change of the low energy physics.