Similar to what James Black said, don't jump straight ahead and rip replace everything to Flash, before you figure out where the bottleneck is (if one exists).
I think you might also be missing some parts of the picture, judging from some of the comments...
I want to lay out all the layers in your communications, just to make things clear - point is, its unlikely ALL of it is the problem, and youd be better looking at the problematic layer only.
- Client code - currently javascript, which calls the next layer asynchronously. If you optimize to Flash, this part may be more responsive, and its "richer".
- TCP/IP (wont go lower than that) - this is ALWAYS stateful, because TCP is stateful. At least at the connection layer... What this means, is that usually the TCP connection is kept open for a long time, and you dont open a new one on each request. You won't be changing this in a browser app...
- HTTP - stateless in principal, but typically circumvented through some form of cookies and server-side session. Also it's not the MOST efficient protocol, especially for binary data, since it is text-based and a bunch of overhead. While it is technically possible to skip through this protocol, its highly UNrecommended to do this in a browser app, because (a) its unexpected from a user point of view, and (b) its not very firewall friendly.
- XML - if you discover that your bottleneck is in the amount of data transmitted, you might just want to switch out the payload format since XML is pretty verbose. For example, JSON would be a great alternative here. Or maybe just trim down the XML schema...
- Server side app - often, the bottleneck will be here, regardless of anything that happens downstream.
So, to sum up - switching your client to Flash might have 2 possible benefits: the client itself may run faster (depending on your client), and it allows you to call sockets directly bypassing 3 above (http). Again, note that the 2nd benefit is dubious at best - benefit is questionable, and there are clear downsides to it.
Unless the bottleneck is the client display code, you're better off either switching to JSON (or other data format), or optimizing your server code. Once you profile and figure out where the problem is, you'll better know where to focus your efforts. I find it highly unlikely that Flash will help with that. (again, since it IS a game, you may need the improved display).