I get nothing when I:
rpm -qa | grep dovecot ps aux | grep dovecot service dovecot status
I suppose you are expecting that after doing a "make install", the application already runs, like you are used from rpm installs ? That will not be the case. I am not even sure if the dovecot install process places a start/stop script in /etc/init.d. Often you have to do many things on your own after a source install, like adapting or creating a start/stop script or copying and modifying a sample configuration file. That's what Mark pointed out : you should probably at least go through the provided configuration file and then you can try and start dovecot by hand and observer the relevant logfiles, depending on your dsitribution. If you find that all works well, you can use the start/stop mechanism to integrate dovecot startup into your system startup. Sorry, but when compiling source packages you have to "RTFM" and adapt the explanations to your configuration.
If you configured dovecot and started it and it fails to run, start it from the command line and watch out for fatal errors on STDOUT or in the logs. Most probably the ports are in use by another pop/imap program.
Hope this helps, Jakob Curdes