Boron-nitride graphene-like monolayer possesses a similar atomic arrangement to that of the famous graphene. However, due to the large difference in electronegetivity between boron and nitrogen atoms, the electronic properties of the two nanomaterials are different significantly. Here, we report on our theoretical investigation of the electronic structure and spin-polarization of zigzag-edged boron-nitride triangular nanoflake using a Hubbard model and the first-principles calculations within density-functional theory. Our numerical results indicate that in contrast to graphene nanoflake with spin-polarized ground state, the boron-nitride nanoflak has the zero-energy state that is either empty or fully occupied, and its ground state is thus spin-unpolarized which breaks the Lieb's law. However, the electron occupation and spin-polarization of the zero-energy state of boron-nitride nanoflake can be tuned by doping it with electrons or holes. These results are expected to offer the theoretical basis for the applications of boron-nitride nanomaterials in spintronics.