Bathtub drains that do not have mechanical stoppers usually lack the cross-hairs that keep large objects, such as washcloths, from falling inside the drain. Since the bathtub drain is often smaller than the main house drain, an object the size of a washcloth is likely to stop before getting to the main line. Removing a washcloth from a bathtub drain is a matter of getting a good hold on it and pulling it back out. Pulling the washcloth out of the drain, rather than pushing it through, prevents it from clogging in a place that requires the disassembly of your house drain lines.
Reach inside the drain with a pair of long tweezers or a pair of hemostats. Try to grab any small edge of the washcloth and pull it straight out. If you lose the grip, grab the washcloth again.
Continue pulling until part of the washcloth is above the drain opening. Grab the washcloth with your fingers to remove it completely from the drain.
Insert the coil end of a drain snake into the tub drain if you cannot reach the washcloth with simple tools. Do not use a toilet auger. Drain snakes have coils on the ends that break up clogs. The coil will also snag the cotton fabric of a washcloth.
Rotate the handle of the drain snake until you feel the snake reach the washcloth. There will be resistance when you reach the washcloth. Rotate the snake handle clockwise another full turn to "dig" the coil into the fabric.
Turn the snake handle counterclockwise one full turn rapidly. This will usually snag the fabric onto the coil end. Pull the snake out of the drain slowly until you can reach the washcloth with your fingers. Repeat the snagging process as often as necessary to pull the washcloth out of the drain.