We present high-resolution (1".0) Atacama Large Millimeter/submillimeter Array (ALMA) observations of CO (1-0) and CO (2- 1) rotational transitions toward the nearby IR-luminous merger NGC 1614 supplemented with ALMA archival data of CO (3-2), and CO (6-5) transitions. The CO (6-5) emission arises from the starburst ring (central 590 pc in radius), while the lower-$J$ CO lines are distributed over the outer disk ($∼$ 3.3 kpc in radius). Radiative transfer and photon dominated region (PDR) modeling reveal that the starburst ring has a single warmer gas component with more intense far-ultraviolet radiation field ($n_{\rm{H_2}}$ $∼$ 10$^{4.6}$ cm$^{-3}$, $T_{\rm{kin}}$ $∼$ 42 K, and $G_{\rm{0}}$ $∼$ 10$^{2.7}$) relative to the outer disk ($n_{\rm{H_2}}$ $∼$ 10$^{5.1}$ cm$^{-3}$, $T_{\rm{kin}}$ $∼$ 22 K, and $G_{\rm{0}}$ $∼$ 10$^{0.9}$). A two-phase molecular interstellar medium with a warm and cold ($>$ 70 K and $∼$ 19 K) component is also an applicable model for the starburst ring. A possible source for heating the warm gas component is mechanical heating due to stellar feedback rather than PDR. Furthermore, we find evidence for non-circular motions along the north-south optical bar in the lower-$J$ CO images, suggesting a cold gas inflow. We suggest that star formation in the starburst ring is sustained by the bar-driven cold gas inflow, and starburst activities radiatively and mechanically power the CO excitation. The absence of a bright active galactic nucleus can be explained by a scenario that cold gas accumulating on the starburst ring is exhausted as the fuel for star formation, or is launched as an outflow before being able to feed to the nucleus. ; Comment: 20 pages, 19 figures, 2 tables, accepted for publication in ApJ