We have deployed Google Apps for our email so don't use POP/IMAP.
However, some of our website's scripts (WordPress, OpenCart) use PHP mail function to send emails.
Do I need to keep running a POP3/IMAP server like Dovecot running or can I safely turn it off?
On that note, can I close POP3/IMAP ports safely?
POP/IMAP are for retrieving email. For sending, which is what PHPMailer is doing, you need an SMTP server, e.g. postfix, sendmail, exim.