WooCommerce – autocomplete downloadable orders
When a downloadable order is made in WordPress/WooCommerce 2.1 the order is set to ‘processing’ the same as an order that is physically shipped. The order can then be marked as ‘complete’ by the shop admin, but it makes more sense for the order to be completed as soon as it’s paid for. Luckily, there is a plugin to take care of that: https://wordpress.org/plugins/woocommerce-autocomplete-order/
The product needs to be marked as both ‘virtual’ and ‘download’ for the plugin to automatically mark orders as complete. The Paypal IPN settings mentioned in that article were not required in my case.