With the use of an ordinary household product, Aaron Marino of alpha m. demonstrates how to remove deodorant marks from dark colored shirt. Not getting the marks in the first place is the best recommendation.
Prevention
Put your shirt on first, then pull out your shirt and put on the deodorant by going up the shirt. You can also put the deodorant on and use a hairdryer to dry the product. You could also use a non-chalky product.
Removal
But, if you do get deodorant on your shirt, use a used dryer sheet. Wad it up and rub. Magically disappears!