I use a load balancer with my web site. The browser initiates a websocket connection to my app server. Does the open connection consume any resources on the LB or is it dire
It depends!
The most efficient load balancers listen for requests, do some analysis, then forward the requests; all the bits do not travel through the load balancer. The network forwarding happens at a lower network layer than http (e.g., it is not an http 302 redirect - the client never knows it happened, maintaining privacy around internal network configuration - this happens at OSI Level 4 I think).
However, some load balancers add more features, like acting as SSL endpoints or applying gzip compression. In these cases, they are processing bits as they pass through (encrypt/decrypt or compress in this case).
A picture may help. Compare the first diagram with the second & third here, noting redirection in the first that is absent in the others.