Can't log in to Drupal instance after duplicating database
First check whether $cookie_domain is set to localhost or not?

If it is set as the URL of the live site then it might not work. I faced this issue previously.

Other cause might be clean urls. You can check whether clean URL is enabled in your localhost or not.

You can check whether mod_rewrite is on or not here. If clean url is causing problem then you can disable from your database. If you want to disable clean url goto your database find variables table, find clean_url variable and set it to 0.