Unlike 'select' and other form elements, validation handlers will run when applying '#ajax' to a submit or button element.
To bypass validation, make the element type button, and set empty array values for #limit_validation_errors and #submit.
$form['my_ajax_button'] = array(
'#type' => 'button',
'#name' => 'my_ajax_button',
'#value' => t(BUTTON_FORGOT_YOUR_PASSWORD),'#ajax' => array(
'callback' => 'zuser_ajax_triprequest_user_form_toggle',
'wrapper' => 'user_fields_replace'
),
'#limit_validation_errors' => array(),
'#submit' => array(),
);
Note: You can then use $form_state['triggering_element']['#name'] within your form to detect when this button was clicked.