what sort of thing is likely to cause my game (ran as server using -batchmode flag) to consume the entire CPU after about 8 hours of running?

Start by checking the machine's memory usage at that point. Is it significantly higher than when you started the process? How's the pagefile usage looking?

"Surely X can't be the culprit?" are famous last words. For instance, your logging probably does a lot of string concatenation and formatting. If you're doing that sub-optimally - for instance, using the + operator instead of a stringbuilder and string.Format - the resulting allocation and destruction of strings could have an effect on Mono's less-than-ideal garbage collector (see these two articles). While StringBuilder is not always faster, it's memory fragmentation that you should care about the most.

While I'm not saying that logging is the culprit, it's a good example of things that might be non-obvious at first but where you want to start checking your memory allocation and usage patterns.