How can I check if a var is equal a translated label?

by

You need to pass the text domain too. Assuming that you have the proper .mo and .po file, and loaded the text domain successfully:

 if( $_POST["save"] === __( 'Publish', 'text-domain' ) ){
    //do something
 }

But this is not generally a good idea. You should do your checks based on something more certain, not a string that can be changed anytime.