Thermally induced domain wall motion in a magnetic insulator was observed using spatiotemporally resolved polar magneto-optical Kerr effect microscopy. The following results were found: (i) the domain wall moves towards hot regime; (ii) a threshold temperature gradient (5 K/mm), i.e., a minimal temperature gradient required to induce domain wall motion; (iii) a finite domain wall velocity outside of the region with a temperature gradient, slowly decreasing as a function of distance, which is interpreted to result from the penetration of a magnonic current into the constant temperature region; and (iv) a linear dependence of the average domain wall velocity on temperature gradient, beyond a threshold thermal bias. Our observations can be qualitatively explained using a magnonic spin transfer torque mechanism, which suggests the utility of magnonic spin transfer torque for controlling magnetization dynamics.