I believe Burp is intended to be used on a single machine for web app testing.  Configure the browser on your computer that's running Burp, not a remote machine.  It sounds like you have Burp setup, but instead of configuring the proxy settings on your iPhone, configure the proxy settings in the web browser on your computer running Burp.  Then you browse to a website using the same machine, capturing all the HTTP traffic and do with it as you please.  The proxying happens all on the same computer instead of happening across multiple computers.

You need to turn off the "Loopback only" proxy listener option in Burp, then you can point any remote device at the burp proxy IP/port, and it'll work just like any other proxy.  Burp defaults to "loopback only" (ie local local machine interface) so the proxy does not accept remote incoming connections from random machines.