Abstract: We study the effect of dark matter (DM) particles in the Sun, focusing, in particular, on the possible reduction of the solar neutrinos flux due to the energy carried away by DM particles from the innermost regions of the Sun, and to the consequent reduction of the temperature of the solar core. We find that in the very low-mass range between 4 and 10 GeV, recently advocated to explain the findings of the DAMA and CoGent experiments, the effects on neutrino fluxes are detectable only for DM models with a very small, or vanishing, self-annihilation cross section, such as the so-called asymmetric DM models, and we study the combination of DM masses and spin-dependent cross sections which can be excluded with current solar neutrino data. Finally, we revisit the recent claim that DM models with large self-interacting cross sections can lead to a modification of the position of the convective zone, alleviating or solving the solar composition problem. We show that when the "geometric'' upper limit on the capture rate is correctly taken into account, the effects of DM are reduced by orders of magnitude, and the position of the convective zone remains unchanged.