What specifically is incorrect about the Dirac Sea interpretation?

The Dirac sea interpretation cannot deal with bosonic antiparticles. At the time it was conceived, I don't think physicists were aware that antibosons existed. A simple example of a boson-antiboson pair are the $W^\pm$ bosons. In order to prevent an electron from falling into a negative energy state, we use the Pauli principle. A fermion cannot fall into a fully occupied sea of other fermions. This obviously does not apply to bosons.