This sounds like an interesting problem. From reading the various comments, it sounds like you want this to work when you're on a local network - so you have wifi, but the wifi router/base station isn't connected to the actual internet?

Because background refresh isn't going to be predictable - you'll never know when it is going to update - you might want to get creative.

You could look into exploiting iOS VOIP support, only without the Voice! Apple has some tips on VOIP here. VOIP basically uses something called SIP (Session Initiation Protocol), which is signalling layer of the call, and a lot like HTTP. It's this SIP layer that you want to take advantage of.

This isn't going to be terribly easy, but it should be achievable. Setup your app to use VOIP, and then look into something like PJSip as your SIP library. Then, on your local network have a SIP Server (I'm sure there are plenty open source implementations) that you can register your iPhone against (so your server knows where your phone is, pretending to be a VOIP phone). This should work, because it doesn't need to go through Apple as far as I am aware... And will run happily on your local network.

Then, the server can send a message via SIP to the handset, as if it were instigating a VOIP session. You app is awoken, gets the messages - ideally from the SIP message if possible - and then just doesn't start the session. SIP was designed just for creating sessions, not just VOIP. When I worked in Telecoms R&D (a long time ago) we were using it to swap between Text/Voice/Video, all using local servers.

You'll have to jump a lot of hoops to make this work, but it would be pretty awesome. I have never tried this actual use case - especially with iOS, but I'm fairly sure it will work. It is a bit of a fudge, but should get you where you need to go.

Good luck!