Low-temperature dissipation of mechanical energy is studied in AlGaAs/GaAs-based nanomechanical resonators with a two-dimensional electron gas. It is experimentally shown that the temperature dependence of dissipation demonstrates a peak near 30 K. A short illumination leads to a persistent change in the quality factor, which can be compared with the persistent photoconductivity effect. In particular, the illumination persistently suppresses the dissipation peak. This suppression shows that the nature of the peak is not related to the thermoelastic and Akhiezer damping. A hypothesis associating the peak with DX-centers or similar low-symmetry and light-sensitive centers is proposed. The observed effects should be taken into account when studying the low-temperature dissipation using optical methods.