Here's Apple's soup-to-nuts troubleshooting page for Reminders and Calendar.
The key for this issue is that Reminders is linked to Calendar.
On the Mac, Calendar has a Refresh item in the View menu. Choosing that refreshes Reminders as well.
On iOS, it's similar: Tap Calendars in the top-left corner to load the Calendars list. Tap the Refresh button in the lower-left corner of the Calendars list.
This could also be important in certain locations:
Open a secure website from your iOS device to test if you are online as is necessary for iCloud Reminders. This also tests if the main ports 80 and 443 are accessible. Reminders requires port 443 access to iCloud in order to push and pull updates to and from the iCloud Calendar servers.