  We present the velocity dispersion and dynamical mass estimates for 270
galaxy clusters included in the first Planck Sunyaev-Zeldovich (SZ) source
catalogue, the PSZ1. Part of the results presented here were achieved during a
two-year observational program, the ITP, developed at the Roque de los
Muchachos Observatory (La Palma, Spain). In the ITP we carried out a systematic
optical follow-up campaign of all the 212 unidentified PSZ1 sources in the
northern sky that have a declination above $-15^\circ$ and are without known
counterparts at the time of the publication of the catalogue. We present for
the first time the velocity dispersion and dynamical mass of 58 of these ITP
PSZ1 clusters, plus 35 newly discovered clusters that are not associated with
the PSZ1 catalogue. Using Sloan Digital Sky Survey (SDSS) archival data, we
extend this sample, including 212 already confirmed PSZ1 clusters in the
northern sky. Using a subset of 207 of these galaxy clusters, we constrained
the $M_{\rm SZ}$--$M_{\rm dyn}$ scaling relation, finding a mass bias of $(1-B)
= 0.83\pm0.07$(stat)$\pm0.02$(sys). We show that this value is consistent with
other results in the literature that were obtained with different methods
(X-ray, dynamical masses, or weak-lensing mass proxies). This result cannot
dissolve the tension between primordial cosmic microwave background
anisotropies and cluster number counts in the $\Omega_{\rm M}$--$\sigma_8$
plane.
