Probably not. "Steal the shirt off someone's back" means to take everything from a person, even the most basic and least significant item of clothing that they are wearing, and leave them with nothing.
If you give someone a $20 shirt, causing yourself a little embarassment, that's not quite the same commitment to charity as giving them your $20,000 car.
Think of it this way; my shirt would have a lot more value to me than value to you. Therefore, if I gave you the shirt off my own back would show a high degree of selflessness.
Also probably the source of the "they'd take the shirt of his back" saying.