Drupal - Removing the upload button in managed files

The work to separate out the managed_file element is done in file_managed_file_process(), which is the default #process function for that type of element.

To hide the upload button you need to implement your own simple #process function for the element:

function MYMODULE_my_form($form, &$form_state) {
  $form['image'] = array(
    '#type' => 'managed_file',
    '#title' => 'File',
    '#upload_location' => 'public://my-files/',
    '#process' => array('MYMODULE_my_file_element_process')
  );

  return $form;
}

function MYMODULE_my_file_element_process($element, &$form_state, $form) {
  $element = file_managed_file_process($element, $form_state, $form);
  $element['upload_button']['#access'] = FALSE;
  return $element;
}