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It is more likely that the spots are from your washing machine re: oil is leaking out of the transmission.
They will be almost impossible to see when the clothes are wet.

There are a couple ways to check this.

1.
Run a load without soap in the washer.
Use the same cycle and water temperature that produces the spots.
Stop the unit close to the end of the wash cycle with the unit full of water.
Let it sit for about half an hour.
Then with a flashlight look for small oil droplets floating on top of the water.

2.
Take clothes or what ever type of cloth that is spotting.
Make sure they have no spots.
Wet them in the sink and wring them out by hand.
Then dry them in the dryer.
If there are no spots after drying odds are that the washer is the culprit.

I am not sure that anything will remove the spots once the stain has been set in the dryer.
You could try mineral spirits or varsol but be sure to test this first on a piece of clothing that you are not fond of as it may ruin the item.
You also have to be careful when using a solvent. Do not place the item in the dryer until all the vapors have evaporated as they are volatile.