We explore the heat current in the quantum Hall edge at filling factors \nu = 1 and \nu = 2 in the presence of dissipation. Dissipation arises in the compressible strip forming at the edge in presence of a smooth confining potential. Such strip was predicted to host an infinite number of hydrodynamic neutral modes, which however were never observed. A possible explanation may be in their dissipative nature, which was not fully considered before. Heat transport measurements are capable of detecting neutral modes and experiment [H. le Sueur et al., Phys. Rev. Lett. 105, 056803 (2010)] at \nu = 2 captured additional degrees of freedom transferring heat at the edge. Surprisingly, the breakdown of heat current quantization has been found. We conjecture that the aforementioned dissipative modes might be responsible for this behavior. We build a low-energy effective model and show that the lowest hydrodynamic mode carries a portion of the heat flux quantum which is the same both at \nu = 1 and \nu = 2. Although our results are consistent with the experiment, a microscopic model of dissipation is needed to confirm the prediction of the low-energy approximation.