If your feed has previously been displaying just fine then randomly seems to stop working, it is most likely due to your access token expiring.
Version 6.0 and above
You can get a new access token by going to WordPress Dashboard > Instagram Feed > Settings, clicking the red trashcan icon to delete your current source, and then clicking Add Source to reconnect your account.
Below version 6.0
You can get a new access token by using the big blue button on the “Configure” tab.
We’ve received mixed reports of the frequency of the expiration of Instagram Access Tokens. Most tokens very rarely expire and never need to be renewed, but a few users have reported their token expiring every few days. All that Instagram’s documentation says is “Access tokens may expire at any time in the future.”.
As of version 2.0 of the Free plugin and 4.0 of the Pro plugin, you can connect as a “business” account to the new API. These access tokens seem to rarely, if ever, expire. See this FAQ for more information.