Using the tensor network approach, we investigate the monomer-dimer models on a checkerboard lattice, in which there are interactions (with strength v) between the parallel dimers on half of the plaquettes. For the fully packed interacting dimer model, we observe a Kosterlitz-Thouless (KT) transition between the low-temperature symmetry breaking and the high-temperature critical phases; for the doped monomer-dimer case with finite chemical potential $μ$, we also find an order-disorder phase transition which is of second order instead. We use the boundary matrix product state approach to detect the KT and second-order phase transitions and obtain the phase diagrams v-T and $μ-$T. Moreover, for the noninteracting monomer-dimer model (setting $μ$=$ν$=0), we get an extraordinarily accurate determination of the free energy per site (negative of the monomer-dimer constant h2) as f=-0.662798972833746 with the dimer density n=0.638123109228547, both of 15 correct digits.