Drupal - How to a use a template file to theme a form?

It's completely reasonable to want to use a tpl file to display a form. You can use lots of extraneous CSS and #prefix/#suffix properties to achieve similar results, but by using tpl's you don't have to clutter up the separation of your logic and presentation layers and don't have to target ugly CSS selectors like #user-login label. Here's an example in Drupal 7...

mytheme/template.php:

function mytheme_theme($existing, $type, $theme, $path) {
    // Ex 1: the "story" node edit form.
    $items['story_node_form'] = array(
        'render element' => 'form',
        'template' => 'node-edit--story',
        'path' => drupal_get_path('theme', 'mytheme') . '/template/form',
    );

    // Ex 2: a custom form that comes from a custom module's "custom_donate_form()" function.
    $items['custom_donate_form'] = array(
        'render element' => 'form',
        'template' => 'donate',
        'path' => drupal_get_path('theme', 'mytheme') . '/template/form',
    );

    return $items;
}

custom_donate_form():

function custom_donate_form($form, &$form_state) {
    $form['first_name'] = array(
        '#type' => 'textfield',
        '#attributes' => array('placeholder' => t('First name')),
    );
    $form['last_name'] = array(
        '#type' => 'textfield',
        '#attributes' => array('placeholder' => t('Last name')),
    );
    $form['address'] = array(
        '#type' => 'textfield',
        '#attributes' => array('placeholder' => t('Address')),
    );
    $form['city'] = array(
        '#type' => 'textfield',
        '#attributes' => array('placeholder' => t('City')),
    );
    $form['state'] = array(
        '#type' => 'select',
        '#options' => array(
            'default' => 'State',
            '...' => '...',
        ),
    );
    $form['zip'] = array(
        '#type' => 'textfield',
        '#attributes' => array('placeholder' => t('Zip')),
    );
    $form['email'] = array(
        '#type' => 'textfield',
        '#attributes' => array('placeholder' => t('Email')),
    );
    $form['phone'] = array(
        '#type' => 'textfield',
        '#attributes' => array('placeholder' => t('Phone')),
    );
    $form['submit'] = array(
        '#type' => 'submit',
        '#value' => 'Submit',
    );

    return $form;
}

mytheme/template/form/donate.tpl.php:

<div class="row">
    <div class="small-12 medium-12 large-8 columns">

        <div class="row">
            <div class="small-12 columns">
                <h5>Contact Information</h5>
            </div>
        </div>

        <div class="row">
            <div class="small-12 large-6 medium-6 columns">
                <?php print render($form['first_name']); ?>
            </div>
            <div class="small-12 large-6 medium-6 columns">
                <?php print render($form['last_name']); ?>
            </div>
        </div>

        <div class="row">
            <div class="small-12 medium-6 large-6 columns">
                <?php print render($form['address']); ?>
            </div>

            <div class="small-12 medium-6 large-6 columns">
                <?php print render($form['city']); ?>
            </div>
        </div>

        <div class="row">
            <div class="small-12 medium-3 large-3 columns">
                <?php print render($form['state']); ?>
            </div>

            <div class="small-12 medium-3 large-3 columns">
                <?php print render($form['zip']); ?>
            </div>

            <div class="medium-6 large-6 columns"></div>
        </div>

        <div class="row">
            <div class="small-12 medium-6 large-6 columns">
                <?php print render($form['email']); ?>
            </div>

            <div class="small-12 medium-6 large-6 columns">
                <?php print render($form['phone']); ?>
            </div>
        </div>
    </div>

    <div class="row">
        <div class="small-12 medium-12 large-8 large-offset-2 columns">
            <?php print render($form['submit']); ?>
        </div>
    </div>
</div>

<!-- Render any remaining elements, such as hidden inputs (token, form_id, etc). -->
<?php print drupal_render_children($form); ?>

This is using Foundation, which gives us a form like this:

You have to implement hook_form_alter() in a module or template.php and set the form's #theme property:

/**
 * Implements hook_form_alter().
 */
function hook_form_alter(&$form, &$form_state, $form_id) {
  if ($form_id == 'user_login') {
    $form['#theme'] = array('overwrite_user_login');
  }
}

Then implement new theme:

/**
 * Implements hook_theme().
 */
function hook_theme($existing, $type, $theme, $path){
  return array(
    'overwrite_user_login' => array(
      'render element' => 'form',
      'template' => 'form--user_login',
      'path' => $path . '/templates',
    ),
  );
}

And then add form--user_login.tpl.php template with follow code to render form:

<?php print drupal_render_children($form) ?> 

Even though you may be able to use kiamlaluno's solution I personally wouldn't.

What is your reason for needing a template file for a form? If it's because you want slightly different markup for an existing form? If so then you can use hook_form_alter() to modify the form before it is rendered. Using the Form API you you can modify all the form fields inject html elements etc.

Here is an example of hook_form_alter() that I've created that modifies the standard drupal login form block:

/**
 * Implements hook_form_alter().
 */
function MYMODULE_form_alter(&$form, &$form_state, $form_id) {

  switch ($form_id) {
    case 'user_login_block':

      // Form modification code goes here.
            $form['divstart'] = array(
                '#value' => '<div style="background-color: red;">',
                '#weight' => -1,
            );

            $form['instruct'] = array(
                '#value' => '<p>Enter your username and password to login</p>',
                '#weight' => 0,
            );          

            $form['divend'] = array(
                '#value' => '</div>',
                '#weight' => 4,             
            );
      break;
  }
}

The above example wraps the entire form within a DIV which has an inline style to turn the background colour to red. It also adds a paragraph of help text to the beginning of the form.

This is what my user login form looks like now once the above code is loaded:

See the Form API reference for more information: Form API Reference