The TASTE project is searching for low-mass planets with the Transit Timing Variation (TTV) technique, by gathering high-precision, short-cadence light curves for a selected sample of transiting exoplanets. It has been claimed that the hot Jupiter WASP-3b could be perturbed by a second planet. Presenting eleven new light curves (secured at the IAC80 and UDEM telescopes) and re-analyzing thirty-eight archival light curves in a homogeneous way, we show that new data do not confirm the previously claimed TTV signal. However, we bring evidence that measurements are not consistent with a constant orbital period, though no significant periodicity can be detected. Additional dynamical modeling and follow-up observations are planned to constrain the properties of the perturber or to put upper limits to it. We provide a refined ephemeris for WASP-3b and improved orbital/physical parameters. A contact eclipsing binary, serendipitously discovered among field stars, is reported here for the first time.