Mark the bottom of the door with a marker about 1/2 inch above the carpet line to ensure a smooth swing when you shorten the door. Remove the door from its hinges with a screwdriver and set it over a sawhorse. Draw a straight line across the door and through the mark with a pencil. This is the line you must cut along to properly shorten the door. Put a strip of painter's or masking tape along the line to reduce splintering when you make the cut.
Use a utility knife and cut along the line you drew several times, which is called "scoring." This creates a perforation that makes the door easier to slice through. Set up your circular saw and make sure its blade and teeth are sharp. Slice through the door using the line as a guide until the bottom portion falls off. It should now be short enough to swing freely.
Sand the bottom of the door until it's smooth, then paint it to match the color of the rest of the door. Your closet door may have been hollow, in which case you'll have to fill in the gap with a piece of scrap wood and wood glue. Separate the bottom strip of the portion of the door you sliced off with a utility knife and glue it to the bottom of the door. Clamp it in place until it dries.
Hang the door back up and swing it. It should swing freely without touching the carpet. Repeat the steps with the other closet doors that are affected by the high carpet.