Pull out the drawer and empty its contents.
Check the sides of the drawer, especially the wood channels, for signs of unusual wear. This helps locate the source of the scraping sound. You can also reinsert the drawer, then open and close it until you isolate the scraping.
Stand the drawer on its side to check the construction. If assembled with nails, use the hammer to tap gently on each nail head to tighten the drawer. Use a screwdriver to tighten drawers assembled with screws.
Insert the drawer in the dresser, and slide in and out to listen for any further scraping.
Remove the drawer and use the sander to plane the sides where the drawer scrapes inside the dresser. Move the sander in only one direction for consistency; push the sander along the side of the drawer, then lift the sander and return it to the starting position.
Wipe away excess sawdust and test the drawer in the dresser.
Rub the end of a wax candle on the wood areas that scrape if you still hear a sound. Wax helps lubricate the parts to eliminate the noise.