Drop ceiling tiles can become stained and ruined from the condensation emitted by an air conditioning system. To solve the problem you need to insulate the duct work that is sweating and causing the staining.
Waterproof Wrap Insulation
This type of insulation is easy to install and contains no fiberglass. You can purchase the insulation from your local hardware store. Wrap the insulation tightly around the duct work. Secure the wrap insulation with foil back tape because it will last forever. Be sure you use enough waterproof wrap insulation to prevent any further moisture which could lead to mildew and mold.
You may want to speak with a professional heating and cooling service person about getting the best wrap insulation for the problem. You might also want to get some help in wrapping the insulation around the duct work. The service person might have some advice about other solutions for your situation.
Replace the Tiles
Be sure that your insulation solution is working before you tackle the job of fixing the tiles. The stained tiles can be coated and then painted if replacement is not an option. Otherwise, remove the stained tiles and take one of them to the local hardware store to purchase replacement tiles that match the existing ones.