jmc

Checking the primes less than $\sqrt{67}$, namely 2, 3, 5, and 7, as potential divisors, we find that 67 is prime. Thus, $\frac{67}{2x-23}$ is an integer if and only if $2x-23=\pm 1$ or $2x-23=\pm 67$. The first equation yields $x=12$ or $x=11$ and the second gives $x=45$ or $x=-22$. The sum is $12+11+45-22=\boxed{46}$.