Will I Be Logged Out If the Connection to an RDS for SQL Server Instance Times Out?

No, you will not be logged out if the connection times out. You can modify the remote query timeout parameter to adjust how long a remote operation can take in seconds before it times out. The default value is 600, which is a 10-minute wait. This value applies to an outgoing connection initiated by the DB engine as a remote query. This value has no effect on queries received by the DB engine. For details about how to modify parameters, see Modifying Parameters.