Zend Framework 2 limit REST API to accept only application/json or application/xml

You can connect a listener to onroute event, check the Accept header value and return a 406 Not Acceptable response for all headers other then application/json or application/xml.

In onBootstrap connect your listener:

$eventManager->attach($serviceManager->get('Application\Listener\RestAcceptListener'));

In your listener check the Accept header

/**
 * Check Accept header
 *
 * @param MvcEvent $event
 * @return Response
 */
public function onRoute(MvcEvent $event)
{
    $routeMatch = $event->getRouteMatch();
    $controller = $routeMatch->getParam('controller');

    // To limit for rest calls only you can do some controller check here
    // You can also do instanceof check this is all up to you...
    if( $controller !== 'somecontroller'){
        return;
    }

    $request = $event->getRequest();
    $headers = $request->getHeaders();

    $acceptHeader = $headers->get('Accept');

    // Check whether accept type corresponds to the allowed ones
    $accept = array('application/json', 'application/xml');
    if(!$acceptHeader->match($accept)){
        $response = new Response();
        $response->setStatusCode(406);
        return $response;
    }
}

UPDATE:

To make a module check you can use namespace of the controller. For example to check Application module using php explode:

$parts = explode('\\', $controller, 2);
if ($parts[0] !== 'Application'){
    // We do not have a controller from Application module
    return;
}