In this work, we demonstrate the optical heating modulation of soliton-based supercontinuum generation through the employment of multi-walled carbon nanotubes (MWCNTs) acting as fast and efficient heat generators. By utilizing highly dispersion-sensitive liquid-core fibers in combination with MW-CNTs coated to the outer wall of the fiber, spectral tuning of dispersive waves with response times below one second via exploiting the strong thermo-optic response of the core liquid was achieved. Local illumination of the MW-CNTs coated fiber at selected points allowed modulation of the waveguide dispersion, thus controlling the soliton fission process. Experimentally, a spectral shift of the two dispersive waves towards the region of anomalous dispersion was observed at increasing temperatures. The presented tuning concept shows great potential in the context of nonlinear photonics, as complex and dynamically reconfigurable dispersion profiles can be generated by using structured light fields. This allows investigating nonlinear frequency conversion processes under unconventional conditions, and realizing nonlinear light sources that are reconfigurable quickly.