I use the Reminders app for my grocery lists. When I add items from my computer, or my husband adds things to our shared lists, they don't update on my iPhone for quite a while and I have to refresh them manually. Here's how to force an update of your Reminders on an iPhone, iPad, or iPod Touch:
Open the Calendars app on your iPhone.
Tap the Calendars button at the bottom of the screen.
Pull down on the list of Calendar accounts to refresh your accounts. This will refresh your calendars, which includes updating Notes and Reminders.