What are the things to check to make sure my server's SMTP can be used by PHPMailer similar to that when I'm using GMail's SMTP?
Use this site to verify your server is ok as seen from the outside. From the inside try testing it with a tool like "swaks" or using simple telnet. Does your php application use username/password to send the email ? Does your postfix server require authentication to send email from internal network ip's ? You can try sending an email with :
Watch read what the server responds with carefully :) If you can send email with telnet or swaks without authentication, chances are phpmailer can send too. If not, setup phpmailer to use a user/pass to send the emails.
I would go for this list:
DO YOU HAVE AN OPEN RELAY?
This is really critical, if you do, most of the mailservers will drop you mail.
Is your virus scanner working? This test sends an email with the EICAR virus test file. This is not a virus but a test which triggers all popular virus scanners. The site will send the test file in an email to the address you enter.
Are you on rbl? A link to OpenRBL.org.
Check if you're listed on the : DSLB LISTING, SPEWS.ORG, SPAMHAUS
Do you support client authentication? It is good to see who wants to send that mail.
Also worth to use smtp-s even if it is overhead.