So after using fiddler to watch the request/responses from my app, I am pretty sure the issue is with the route.solve task that gets sent to ArcGIS online.
When I use Firefox and Chrome I can see that this is sent as a GET request and returns in less than 1 second. However, when using IE7 this never even shows up in Fiddler and the map gets stuck. The previous request (a spatial query to find the parks within a 1 mile buffer) is sent via POST to my server but I can see the response so I believe it returned fine.
Any ideas on what could cause an issue with the route.solve task in just IE7?