What port is used by SQL Server to write backups to a remote server?
Remote file access occurs over SMB. Remote backups are just another form of remote file access. See File and Sharing Protocol Stack.

If you use SMB over TCP directly (ie. disable NetBIOS) then the port used is 445, both UDP and TCP. If you use NetBIOS then it requires 137 both UDP and TCP, 138 UDP and 139 TCP. See Directly Hosting SMB over TCP/IP:


  NetBIOS over TCP traditionally uses the following ports:

nbname            137/UDP
nbname            137/TCP
nbdatagram        138/UDP
nbsession         139/TCP

  
  Direct hosted "NetBIOS-less" SMB traffic uses port 445 (TCP and UDP).