Multiple field API example including add/remove submit and callback functions.

addmore.module

<?php

/**
 * @file
 * Module file for add_more_example.
 */

/**
 * Implements hook_menu().
 *
 * @inheritdoc
 */
function addmore_menu() {
  $items = array();
  $items['admin/snippet/addmore'] = array(
    'title' => 'Example form',
    'type' => MENU_CALLBACK,
    'page callback' => 'drupal_get_form',
    'page arguments' => array('addmore_form'),
    'access callback' => TRUE,
  );
  return $items;
}

function addmore_form($form, &$form_state) {
  $form['field_container'] = [
    '#type' => 'container',
    '#weight' => 80,
    '#tree' => TRUE,
    '#prefix' => '<div id="js-ajax-elements-wrapper">',
    '#suffix' => '</div>',
  ];

  $values = variable_get('testvar_one', NULL);
  $form_state['field_deltas'] = isset($form_state['field_deltas']) ? $form_state['field_deltas'] : range(0, count($values) - 1);

  foreach ((array) $form_state['field_deltas'] as $delta => $value) {
    $form['field_container'][$delta] = [
      '#type' => 'container',
      '#attributes' => [
        'class' => ['container-inline'],
      ],
      '#tree' => TRUE,
    ];

    $form['field_container'][$delta]['field1'] = [
      '#type' => 'textfield',
      '#title' => t('Field 1 - ' . $delta),
      '#value' => $values[$delta],
      '#size' => 10,
    ];

    $form['field_container'][$delta]['remove_name'] = [
      '#type' => 'submit',
      '#value' => t('Remove'),
      '#submit' => ['addmore_form_remove'],
      '#ajax' => [
        'callback' => 'addmore_form_remove_callback',
        'wrapper' => 'js-ajax-elements-wrapper',
      ],
      '#weight' => 50,
      '#attributes' => [
        'class' => ['button-small'],
      ],
      '#name' => 'remove_name_' . $delta,
    ];
  }

  // Submit buttons.
  $form['field_container']['add_name'] = [
    '#type' => 'submit',
    '#value' => t('Add one more'),
    '#submit' => ['addmore_form_add'],
    '#ajax' => [
      'callback' => 'addmore_form_add_callback',
      'wrapper' => 'js-ajax-elements-wrapper',
    ],
    '#weight' => 100,
  ];

  $form['submit'] = [
    '#type' => 'submit',
    '#value' => t('Submit'),
    '#submit' => ['addmore_submit'],
    '#weight' => 150,
  ];

  $form['other_field'] = [
    '#type' => 'textfield',
    '#title' => t('Other field'),
  ];

  return $form;
}

/**
 * Remove submit function
 */
function addmore_form_remove($form, &$form_state) {
  $delta_remove = $form_state['triggering_element']['#parents'][1];
  $k = array_search($delta_remove, $form_state['field_deltas']);
  unset($form_state['field_deltas'][$k]);

  $form_state['rebuild'] = TRUE;
  drupal_get_messages();
}
/**
 * Remove callback.
 */
function addmore_form_remove_callback($form, &$form_state) {
  return $form['field_container'];
}

/**
 * Add submit function.
 */
function addmore_form_add($form, &$form_state) {
  $form_state['field_deltas'][] = count($form_state['field_deltas']) > 0 ? max($form_state['field_deltas']) + 1 : 0;
  $fs['field_deltas'] = $form_state['field_deltas'];
  $fs['field_deltas'][] = count($form_state['field_deltas']) > 0 ? max($form_state['field_deltas']) + 1 : 0;

  $form_state['rebuild'] = TRUE;
  drupal_get_messages();
}

/**
 * Add callback.
 */
function addmore_form_add_callback($form, $form_state) {
  return $form['field_container'];
}

function addmore_submit($form, $form_state) {
  print '';
  variable_set('testvar_one', $form_state['input']['field_container']);
  $x = variable_get('testvar_one', NULL);
  print '';
}