Drupal - Getting the image URL from a field_image on a node

Here is an update version of for this answer after 8.0.x. This base on @joelpittet comment.

{{ file_url(node.field_image.entity.fileuri) }}

I don't know about Twig, but you get the file URI with $node->field_image->entity->getFileUri() in PHP, and the you need to call file_create_url() on that:

file_create_url($node->field_image->entity->getFileUri())

I don't know if that is possible in twig. If it's not possible, you can always calculate it in preprocess and add as a new variable.

If someone needed a styled image:

$styled_image_url = ImageStyle::load('large')->buildUrl($node->field_image->entity->getFileUri());