Two-dimensional atomic sheets such as graphene and boron nitride monolayers represent a new class of nanostructured materials for a variety of applications. However, the intrinsic electronic structure of graphene and h-BN atomic sheets limits their direct application in electronic devices. By first-principles density functional theory calculations we demonstrate that band gap of zigzag BN nanoribbons can be significantly tuned under uniaxial tensile strain. The unexpected sensitivity of band gap results from reduced orbital hybridization upon elastic strain. Furthermore, sizable dipole moment and piezoelectric effect are found in these ribbons owing to structural asymmetry and hydrogen passivation. This will offer new opportunities to optimize two-dimensional nanoribbons for applications such as electronic, piezoelectric, photovoltaic, and opto-electronic devices.