The basic answer is no: the user on the client computer (B) can run commands on the server computer (A) (therefore the administrator on the client can run commands on the server), but the administrator on the server can only influence what the user is doing on the client.
However this does not mean that the client computer is fully protected from the server administrator. A malicious server administrator could insert some attack code in an executable file that the user will copy to the client computer and execute there.
Furthermore, if the user has opened a tunnel in addition to the ssh connection, this may provide another attack vector. For example, X11 tunnels can pretty much allow the server administrator to remotely control X11 applications running locally on the client machine (e.g. by injecting key presses or clipboard data). Openssh provides some protection (see the descriptions of -X, -Y and ForwardX11Trusted in the Openssh client manual) through the X11 SECURITY extension, but these controls do not provide absolute protection (X11 was not designed with application isolation in mind, so you can't really have security without sacrificing functionality).