If you see in your logs:
Jun 4 16:11:18 host imapd: /etc/courier-imap/shared/index: No such file or directory

you need to check perms and ownership on /var/qmail/bin/qmail-queue

# ls -la /var/qmail/bin/qmail-queue
-r-s–x–x 1 root qmail 25828 Apr 26 15:20 /var/qmail/bin/qmail-queue
perms are fine, but ownership is not
# chown qmailq:qmail /var/qmail/bin/qmail-queue
# ls -la /var/qmail/bin/qmail-queue
-r-x–x–x 1 qmailq qmail 25828 Apr 26 15:20 /var/qmail/bin/qmail-queue
[root@host ~]# service qmail restart
Stopping : Starting qmail:

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1. Qmail commands
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