Lets imagine you had a .htaccess authentication like that:

AuthUserFile /var/www/vhosts/ask-sheldon.com/.htpasswd
AuthGroupFile /dev/null
AuthName "Sheldons Secret"
AuthType Basic
<Limit GET>
 require valid-user
</Limit>

… and a corresponding password file, you can deactivate the password prompt on 31.12.2014 at 23:59 automatically with this crontab entry:

# remove htaccess password protection on 31.12.2014
MAILTO = mustang@snsconsulting.de
59 23 31 12 * sed -i 's/require valid-user/#require valid-user/g' /var/www/vhosts/ask-sheldon.com/.htaccess