So I was getting errors like this in syslog:

Jul  6 17:35:53 integrity systemd[1]: Started Dovecot IMAP/POP3 email server.
Jul  6 17:35:53 integrity dovecot[10775]: doveconf: Fatal: Error in configuration file /etc/dovecot/conf.d/10-ssl.conf line 79: ssl_dh: Can't open file /etc/dovecot/dh.pem: No such file or directory
Jul  6 17:35:53 integrity systemd[1]: dovecot.service: Main process exited, code=exited, status=89/n/a
Jul  6 17:35:53 integrity systemd[1]: dovecot.service: Failed with result 'exit-code'.

This failure was affecting other parts of my system (i.e. postfix SASL).
The solution was to generate the dh.pem file:

root@integrity:/etc/dovecot
# openssl dhparam -out dh.pem 4096