The first thing I did was try another CPU board, but that didn't work either. I ended up replacing all the boards with the Mod-8 set, and even that had trouble. Sometimes the Reset button would produce nothing. At other times, garbage characters would be printed on the TTY instead of the familiar "--------" prompt.
After pulling my hair out trying to figure out what was happening, I noticed that if I wiggled the boards a bit, it sometimes fixed the problem. In the end I ended up re-heating all the solder joints to the edge connectors on the back-plane. That seemed to have fixed the problem in most cases.
However, my Mod 80 board refused to work. I tested all the ICs in my TL866II and it turned out that one of my 74123 chips didn't pass. I had seen this before on some other boards. Out of all the old TTL chips I had used, these are the only ones that seemed to come up with errors when tested. I am not sure why those seem to have a higher failure rate than other TTL chips, but the majority of them failed. I had some modern 74HC123 chips floating around, but they didn't seem to work in this application, so I have ordered a batch of new 74LS123 chips. Hopefully they will work. I was able to pull a spare "good" 74123 to at least prove that this was the problem, but I need to put that one back at some point..