The recently discovered high-energy transient Swift J164449.3+573451 (Sw J1644+57) is thought to arise from the tidal disruption of a passing star by a dormant massive black hole. Modeling of the broadband emission suggests the presence of a powerful relativistic jet, which contributes dominantly to the observed X-ray emission. Here we suggest that protons can be accelerated to ultra-high energies by internal shocks occurring in the jets, but their flux is insufficient to account for the observed flux of ultra-high energy cosmic rays. High energy protons can produce ~0.1-10 PeV neutrinos through photomeson interactions with X-ray photons. The large X-ray fluence (7x10^-4 erg cm^-2) and high photopion efficiency, together with the insignificant cooling of secondary mesons, result in bright neutrino emission expected from Sw J1644+57 if the jet composition is matter-dominated. One to several neutrinos may be detected by a Km^3-scale detector from one tidal disruption event similar to Sw J1644+57, thereby providing a powerful probe of the composition of the jets.