Bound states in the continuum (BICs) are located states embedded in the continuous spectrum, therefore, they have infinite quality (Q) factor. Here, we find that many types of BICs including symmetry-protected BIC, accidental BIC and Friedrich-Wintgen BIC can exist in bulk Dirac semimetal (BDS) metasurface. We studied the refractive index sensing performance of the quasi-BIC mode that is close to BIC mode, and found that the quasi-BIC has high refractive index sensing sensitivity and figure-of-merit (FOM) when the external refractive index changes. The effect of Fermi energy of the BDS metasurface on quasi-BIC mode is researched. The FOM of the quasi-BIC increases from 202 to 663 when the Fermi energy is adjusted from 0.2 eV to 1 eV, and the sensing sensitivity maintains at about 158 GHz/RIU. The origin of the improved sensor performance is that the Q factor of quasi-BIC mode can be enhanced by controlling the Fermi energy of BDS, which may expand the applications range of this new class of plasmonic materials.