Can health monitor maintain TCP connection?

Is there a reason the development team is asking this question? Generally you would want a new connection for each monitoring request since it allows the F5 to better reflect how a new client request will look. For example, say a service is disabled on the server that disallows new TCP connections but allows current connections to complete naturally. The F5 would continue using the open connection for the monitor, continue marking the node up/available, but actual forwarded client requests would fail.

As @Kevin.K pointed out, there are alternatives to request/response based monitors, but any solution depends on why the development team wants to see fewer connections. From an application-traffic perspective, you can achieve this functionality by assignign a oneconnect profile to HTTP/HTTPS virtual servers, thus re-using server-side connections (non-monitor) for subsequent client requests.