  We study periodically driven pure Kitaev model and ferromagnetic phase of the
Kitaev-Heisenberg model on the honeycomb lattice by off-resonant linearly and
circularly-polarized lights at zero magnetic field. Using a combination of
linear spin wave and Floquet theories, we show that the effective
time-independent Hamiltonians in the off-resonant regime map onto the
corresponding anisotropic static spin model, plus a tunable photoinduced
magnetic field along the $[111]$ direction, which precipitates Floquet
topological magnons and chiral magnon edge modes. They are tunable by the light
amplitude and polarization. Similarly, we show that the thermal Hall effect
induced by the Berry curvature of the Floquet topological magnons can also be
tuned by the laser field. Our results pave the way for ultrafast manipulation
of topological magnons in irradiated Kitaev magnets, and could play a pivotal
role in the investigation of ultrafast magnon spin current generation in Kitaev
materials.
