The recent extremely bright gamma-ray burst (GRB) from a binary neutron star merger, GRB 230307A, may offer a good probe for the production of GRB neutrinos. Within the constraint from IceCube neutrino nondetection, the limits for key physical parameters of this burst are extracted in different scenarios, including the fireball, Poynting-flux-dominated, and hybrid jet. Different from the former nearby “monsters” and due to its smaller isotropic equivalent radiated energy (E
γ,iso ∼ 4 × 1052 erg), the constraint seems loose if nonthermal neutrinos produced from photomeson interactions are the only consideration. However, a quasi-thermal neutrino emission from hadronuclear processes is constrained in this neutron-rich postmerger environment, and the upper limit of the allowed nucleon loading factor is ∼a few. Based on this, a discussion is presented on the possible prompt emission mechanism and jet composition for GRB 230307A in the context of multimessenger astrophysics. It is worth noting that until now, no GRB neutrinos have been ever detected, even for the two brightest nearby GRBs ever observed (GRB 221009A and GRB 230307A), which have different dissipation mechanisms.