Routing changes break the network connection. If it's buffered and you can reconnect transparently, it might work, but for unbuffered connections (like most web connections are) you'll see increased error rates.
This has been proven to not happen often in the real world.
Who are you? We thought we knew most folks who run TCP Anycast... Anyways, I've run multi-day long TCP streams without issue. Like any sort of anycast, there is a lot of finesse, but if you are logical in your network architecture you will have no issues. In fact, you can re-establish TCP connections across datacenters if desired using tools like pfsync.