Add a "Save" action in local task to a node I would like to add a new action to local task on node.
This new action has to save the current node.

I have drupal 8.8.0 

I know how to add a new action.

But my question is about how to save the current edit node from it if it's possible.

My goal is to replace the "save" button at the bottom of the node on edit mode
Local tasks are the wrong tool for the job. Local tasks are for providing context-sensitive action links (e.g. Edit, Delete, View) around an item (e.g. node/user). Trying to make a local task become a form submission button would involve all kinds of hacks that probably would be more trouble than its worth.

If the goal is to alter the save button's position for UI concerns, look at using hook_form_alter to alter the appearance/positioning of the node form's Save button. For example, look at a module like Floating Action Buttons that uses a form alter to place the action buttons in a fixed location.