Solved with RM1729, Rawlat_vanak, mxlcv. Let the total number of tilings of the $8 \times 8$ grid be $N$. For any non-boundary unit segment, the number written on it will be $N - x$, where $x$ is the number of tilings that include a domino across that segment. Since there are $56 + 56 = 112$ such segments, the total value is $112N - \sum x_i$. Now notice that $\sum x_i$ counts $N$ 32 times, so the value of the summation is actually just $80N$ which is divisible by $10$, and so the last digit is zero.