  The neutrino self-energy is calculated in a weakly magnetized plasma consists
of electrons, protons, neutrons and their anti-particles and using this we have
calculated the neutrino effective potential up to order $M^{-4}_W$. In the
absence of magnetic field it reduces to the known result. We have also
calculated explicitly the effective potentials for different backgrounds which
may be helpful in different environments. By considering the mixing of three
active neutrinos in the medium with the magnetic field we have derived the
survival and conversion probabilities of neutrinos from one flavor to another
and also the resonance condition is derived. As an application of the above, we
considered the dense and relativistic plasma of the Gamma-Ray Bursts fireball
through which neutrinos of 5-30 MeV can propagate and depending on the fireball
parameters they may oscillate resonantly or non-resonantly from one flavor to
another. These MeV neutrinos are produced due to stellar collapse or merger
events which trigger the Gamma-Ray Burst. The fireball itself also produces MeV
neutrinos due to electron positron annihilation, inverse beta decay and
nucleonic bremsstrahlung. Using the three neutrino mixing and considering the
best fit values of the neutrino parameters, we found that electron neutrinos
are hard to oscillate to another flavors. On the other hand, the muon neutrinos
and the tau neutrinos oscillate with equal probability to one another, which
depends on the neutrino energy, temperature and size of the fireball.
Comparison of oscillation probabilities with and without magnetic field shows
that, they depend on the neutrino energy and also on the size of the fireball.
By using the resonance condition, we have also estimated the resonance length
of the propagating neutrinos as well as the baryon content of the fireball.
