Do you have a load balancer in front of servers A and B so that you could direct traffic appropriately? Do you have a firewall that may be able to support a rule to route traffice with that "/myApp" to go to B? Those would be my suggestions for ways to do it without changing the servers. However, if you did have to change the server, I would consider having an ISAPI filter that moves the requests to the other machine, but this is likely done better by setting up different domain names so that you have a mywebsite.com and a myapp.website.com so that each can resolve to a different server. If you do that, then server A just has to redirect the request to the other server for a simpler solution. Anyway, that is a few different ways to do it that I can see.
The trade-offs in firewall vs load balancer are really probably quite minuscule as many big sites have load balancers and firewalls, though I'd question the cost difference in having a load balancer versus changing firewall settings. I'd also investigate how feasible each option is for your situation.