You'd really want a backend server to be checking for this. I don't know anything about the instagram API though.
Ideally, the service you would want to monitor would send your backend server an API call whenever this event occurs. Then your backend could do whatever it wanted with this information, such as sending a push notification.
If the service you are monitoring won't send you a REST call though, then things get a bit more complicated. You would want to configure your backend to occasionally (probably every 10-30 minutes) do an API call to Instagram and check to see if there is anything new, and if there is, send it along to the user.
I imagine you could come up with a background task in iOS to do this as well...but I don't think it's a good idea. It would be making continuous API calls in the background quite frequently, and due to Apple's battery restrictions, you cannot always predict when or even if you background tasks would ever actually execute.