Tear away labels are essentially, removable labels. The tear-away labels are beneficial for consumers that do not want the "label itch". In addition, this type of label is beneficial to the fashion industry when companies which to re-label product. When being used by the corporate apparel suppliers, tear away labels can be beneficial to their customers. For example, if a blank t-shirt supplier provides tear away labels, a t-shirt printer or designer can remove the brand label and then sew in or heat transfer onto the t-shirt with the designers own brand label.
One t-shirt supplier that I know has tear away labels is Article1 (http://www.article1.net) Tear away tags or tear away labels from this company or other corporate apparel suppliers, can provide an easy method for a convenient removal of labeling so that other custom labels can be attached.
I am not sure if all of their t-shirts have the tear away label. You would need to reconfirm this point with them directly. In addition, I am not sure if any portion of the label remains visible after the majority is torn away. You would have to research on your own to determine how well the tear away label actually works. I have not personally tried the tear away labeling from the above mentioned company. I only know that they supply this product from one of their newsletters.
Anyway, if you are selling t-shirts and your volume is not large enough to work directly with a factory. The tear away label method may be a good option for you. If you wish to sell t-shirts with your own brand name, it may be a good idea to do further research on the subject. you may want to contact some of the corporate apparel suppliers to find out which of them provide this method of garment labels.