Drupal - How do I disable the "Flood attempts IP blocking" functionality?

What you can do is adding the following code in the settings.php file.

$conf['user_failed_login_ip_limit'] = PHP_INT_MAX;

In this way, the IP will not blocked.

user_login_authenticate_validate() contains the following code.

  if (!empty($form_state['values']['name']) && !empty($password)) {
    // Do not allow any login from the current user's IP if the limit has been
    // reached. Default is 50 failed attempts allowed in one hour. This is
    // independent of the per-user limit to catch attempts from one IP to log
    // in to many different user accounts.  We have a reasonably high limit
    // since there may be only one apparent IP for all users at an institution.
    if (!flood_is_allowed('failed_login_attempt_ip', variable_get('user_failed_login_ip_limit', 50), variable_get('user_failed_login_ip_window', 3600))) {
      $form_state['flood_control_triggered'] = 'ip';
      return;
    }
    $account = db_query("SELECT * FROM {users} WHERE name = :name AND status = 1", array(':name' => $form_state['values']['name']))->fetchObject();
    if ($account) {
      if (variable_get('user_failed_login_identifier_uid_only', FALSE)) {
        // Register flood events based on the uid only, so they apply for any
        // IP address. This is the most secure option.
        $identifier = $account->uid;
      }
      else {
        // The default identifier is a combination of uid and IP address. This
        // is less secure but more resistant to denial-of-service attacks that
        // could lock out all users with public user names.
        $identifier = $account->uid . '-' . ip_address();
      }
      $form_state['flood_control_user_identifier'] = $identifier;

      // Don't allow login if the limit for this user has been reached.
      // Default is to allow 5 failed attempts every 6 hours.
      if (!flood_is_allowed('failed_login_attempt_user', variable_get('user_failed_login_user_limit', 5), variable_get('user_failed_login_user_window', 21600), $identifier)) {
        $form_state['flood_control_triggered'] = 'user';
        return;
      }
    }
    // We are not limited by flood control, so try to authenticate.
    // Set $form_state['uid'] as a flag for user_login_final_validate().
    $form_state['uid'] = user_authenticate($form_state['values']['name'], $password);
  }

The limits are actually two: one for the case Drupal always has an IP, and one for when Drupal has also a user ID. The latter is for the case the user enters a username for an existing account; in that case, Drupal register the user ID, and the IP.
If you want to avoid also that case, then you need to add also this line to the setting.php file.

$conf['user_failed_login_user_limit'] = PHP_INT_MAX;
$conf['user_failed_login_user_window'] = 5;

The Flood Control module will handle this elegantly.

In Drupal 8, you can change the flood settings in the config file user.flood.yml.

uid_only: false
ip_limit: 50
ip_window: 3600
user_limit: 5
user_window: 21600
_core:
  default_config_hash: UYfMzeP1S8jKaaaavxf7nQNe8DsNS-3bc2WSNNXBQWs

This means that per IP and per user, there is a limit:

• Per 3600 seconds (1 hour), 50 attempts per IP address are allowed
• Per 21600 seconds (6 hours), 5 attempts per user account are allowed (quite few)

You might change and import the settings (i set it to 100 attempts per 5 minutes):

uid_only: false
ip_limit: 100
ip_window: 300
user_limit: 100
user_window: 300
_core:
  default_config_hash: UYfMzeP1S8jKaaaavxf7nQNe8DsNS-3bc2WSNNXBQWs