Imagine I have a public display, showing a browser displaying a web page.
Is it possible to send a GET or POST from a mobile device to an HTTP server which triggers some AJAX/pubsub/websocket JavaScript function that changes the page that is presently being viewed on the display, or even just changing an iframe's current domain?
Cross-domain pushstate? Is this at all possible even on your own setup?
You can do this with JavaScript on the client side. This is covered pretty well in that question: Updating address bar with new URL without hash or reloading the page
Unfortunately, it's a relatively new, and mostly unsupported feature. Your alternative is to set the hashtag and use it for navigation.
UPDATE:
If you are trying to "push" pages to the user like a TV channel, then you can have an AJAX request poll the server every few seconds to see if there's a new page. The server would respond with the new URL. You could then put that page in an iFrame.
Assuming you control the web page being shown on the public display, yes.
The web page would need to either periodically contact the server via AJAX, or have a long-running connection to the server (i.e. Comet or WebSockets).
When the server receives the request from the mobile device, it either uses the Comet connection to send the new URL to the web page, or when the web page next contacts it via AJAX, it sends the new URL in response.
The web page then sets its own window.location property to this new URL.
Note that once it's done this, you'll no longer be able to send the browser in question to another new page, unless the page that you've just sent it to also includes the JavaScript that contacts your server.
Then you'd need a browser extension to initiate the connection between the browser and your server.