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(),
);