Context. The relative abundance of the dust grain types in the interstellar medium is directly linked to physical quantities that trace the evolution of galaxies. Because of the poor spatial resolution of the infrared and submillimetre data, we are able to study the dependence of the resolved infrared spectral energy distribution (SED) across regions of the interstellar medium (ISM) with different physical properties in just a few objects. Aims: We aim to study the dust properties of the whole disc of M 33 at spatial scales of 170 pc. This analysis allows us to infer how the relative dust grain abundance changes with the conditions of the ISM, study the existence of a submillimetre excess and look for trends of the gas-to-dust mass ratio (GDR) with other physical properties of the galaxy. Methods: For each pixel in the disc of M 33 we have fitted the infrared SED using a physically motivated dust model that assumes an emissivity index β close to two. We applied a Bayesian statistical method to fit the individual SEDs and derived the best output values from the study of the probability density function of each parameter. We derived the relative amount of the different dust grains in the model, the total dust mass, and the strength of the interstellar radiation field (ISRF) heating the dust at each spatial location. Results: The relative abundance of very small grains tends to increase, and for big grains to decrease, at high values of Hα luminosity. This shows that the dust grains are modified inside the star-forming regions, in agreement with a theoretical framework of dust evolution under different physical conditions. The radial dependence of the GDR is consistent with the shallow metallicity gradient observed in this galaxy. The strength of the ISRF derived in our model correlates with the star formation rate in the galaxy in a pixel by pixel basis. Although this is expected, it is the first time that a correlation between the two quantities has been reported. We have produced a map of submillimetre excess in the 500 μm SPIRE band for the disc of M 33. The excess can be as high as 50% and increases at large galactocentric distances. We further studied the relation of the excess with other physical properties of the galaxy and find that the excess is prominent in zones of diffuse ISM outside the main star-forming regions, where the molecular gas and dust surface density are low.