For example, removing the 6-6, 5-6, and either of 4-6, 5-5, will result in each sum being 19. Removing 1-1, 1-2, and either of 1-3, 2-2, the sum must be 23. (Feel free to prove this.)
I usually do not post problems to which I have no solution, but I have made an exception here. I came within 2 dominoes of a solution, so I feel that one must exist. Please send whatever solutions you have. As always, I will post any unique solutions and credit the solver.