It is sometimes difficult to determine the culprit for grease stains that show up while clothes are drying because there are many sources for grease in laundry. If you put clothes with grease stains in with other clothes, the grease can become loosened in the water and resettle on other clothes. If you are sure that all of your clothes were grease free before the wash, perform some quick checks to identify your problem.
Check the seals around the engine of your dryer for any grease or residue. As with a car's engine, you must lubricate the dryer's engine with oil or grease. As the dryer gets older, the flexible seals around the engine can dry out, allowing grease to leak into the drum and stain clothes. The engine is accessible from the back of most dryers, but you may need to use a screwdriver to remove a protective panel from the back. Ensure that your dryer is off, and run your fingers across as much of the engine and connecting parts as you can reach. If your fingers come back clean, the engine seals are most likely in good condition.
Dryer sheets contain a residue that is intended to stay on your clothes and make them feel softer. If you use too many dryer sheets, the residue from the sheets can build up to the point that it becomes visible. Try reducing the number of sheets you use, especially if you can feel any build-up along the inside of the dryer.
Though actually applied in the washing machine, fabric softener works the same way as dryer sheets to leave a residue on your clothes. If you pour too much fabric softener in with the wash, the fabric softener can become visible when the dryer heats it.
The Washing Machine
Your washing machine can cause grease stains that they are not visible when your clothes are wet. If you can find no problem with your dryer but still find grease stains, inspect the seals in the washing machine, and feel the inside of its drum for any residue. Also ensure that the fabric softener dispenser is working properly if your washer has one.