After you stop smoking cigarettes your throat should begin to heal within a few weeks and usually your throat will tend to get more sore before the sore throat from smoking gets better.
After you stop smoking your throat should be fully healed and stop hurting within 1 month to 2 months.
You can help your throat heal and feel better after smoking by breathing in some steam from a pot of boiling water or using a vaporizer or humidifier or even taking a steamy shower.
Sipping on some warm tea with honey can also help with your hurting throat after giving up smoking.