Not if you don't want to. The "therapeutic" part of therapeutic massage is that you don't have to do anything you are uncomfortable with. Everything that happens during your session should be working to put you at ease, not making you wheeze with fear. Of course, sometimes it is easier and faster to get to certain body parts (such as your low back and hips) if you are not wearing lots of heavy clothing. If you would like to be massaged with clothes on, it's a good idea to wear workout or yoga clothes, so that your muscles are easier to access, and you can be moved around on the table unencumbered.
That being said, even though you are disrobing for your massage, you are not exactly naked. You always have privacy to get dressed and undressed, and once you are undressed, you get right under the massage sheets. Then, you are mostly covered with the sheet (and cozy blanket) for the entire massage. The only body parts being uncovered are the ones that are being worked on. And, you ALWAYS have a right to speak up if something spontaneously bothers you.
Have no fear, there is no need to feel worried about this part (or any part, for that matter!) of your massage session. If you have reservations, I would advise you to just try it and see for yourself how simply "okay" you feel with being disrobed. I've said it before, and I'll say it again: I've seen every body type, every syndrome, and every embarrassing thing you can possibly imagine. So, don't worry about your unshaven legs or that little tummy roll you don't want anyone to see. We're all friends here!
*A note for children receiving massage: the level of clothing depends on the age and the maturity level of the child, and this will be discussed with the parent or guardian of the child upon making the appointment. For most children under the age of 18, parents will be in the room and the child will be massaged with clothes on. In this case, it is recommended that the child be wearing shorts and a tank top, unless they are wiggly babies, and then they may wear a diaper or nothing at all, since the massage style is a bit different, and the parent will be close at hand. As for teenagers, every teen is different. Talking with you as a parent, we may decide together that your 15-year old son comes in for massage on his basketball-injured calf wearing shorts and a T-shirt, while you stay in the room with us. Or your 17-year old daughter who has received many relaxation massages may come in by herself and get undressed as she sees fit. Rest assured, you will be consulted along the way, and your child's best interests will always be first priority.