There are numerous reasons which could cause the problem you report. You do not advise what you see with the iPhone email client. Do you see error reports, or do you just not see new emails appearing in the inbox?
How many mailboxes have you configured in the email application?
Are you seeking to read the new mail under "All Inboxes" or in the in the individual mailboxes?
I have seen issues with one mailbox inhibit retrieving email from other mailboxes, when refreshing under "All Inboxes". Looking at each mailbox in turn and doing a refresh can circumvent such issues until you sort out the failing mailbox.
I have seen the iPhone mail app get its sync status messed up, the only way of fixing that being to delete the mail account and start again. Which iPhone / iOS?