We present a detailed theoretical investigation on strong-field ionization of polar (CO and NO) as well as nonpolar molecules (N 2 , O 2 , and CO 2 ). Our results indicate that accounting for the Stark correction in the molecular tunneling ionization theory leads to overall fairly good agreements with numerical solutions of the time-dependent Schrödinger equation. Furthermore, we show that the effect of dynamic core-electron polarization, in general, has a weak influence on the angle-dependent ionization probability. However, in the case of CO we confirm the recent finding by B. Zhang, J. Yuan, and Z. Zhao [Phys. Rev. Lett. 111, 163001 (2013)] that accounting for dynamic core-polarization is crucial to achieving an overall good agreement with experiments.