I read this to mean:
1. With 24 VDC disconnected from DO, the DI works, and
2. With 24 VDC disconnected from DI, the DO works.
My first guess is that something is shorting the 24 VDC, which has overload protection.
If the situation was either #1 or #2, you would know whether to look for a shorted output or shorted input.
With the situation #1 and #2, the wiring fault depends on where 24 VDC is disconnected (if the problem is a power supply fault), and may be more difficult to find and correct. I suggest closely examining the power supply wiring.
Another option is to pull the removable terminal block (RTB) from each module, one at a time, instead of completely disconnecting 24 VDC. If this also fixed the problem for the other module(s), the problem might be a faulty module itself (the one with its RTB removed when the other(s) work).
Any other ideas?