No, there was nothing in the logs.
I did figure out what the problem was though and how to fix it. I browsed to the /etc/init.d/ folder and then typed ls -l. I noticed that of all the files in that directory, the three services that were giving me problems all had a file size of zero bytes.
I opened each of them up and confirmed that they were empty. I can only believe that something during the apt-get upgrade caused them to be overwritten with empty files.
For each service that wasn't working (courier-imap, courier-imap-ssl, and courier-pop-ssl) I typed apt-get remove --purge [service name] and then did a reinstall. I then checked that the file in /etc/init.d/ was no longer empty, which they weren't. I performed a restart of each service and did some testing on the IMAP server and everything worked as intended.
I posted it all here just in case someone runs across the same problem down the road.