The first thing to understand is that teens are trying very hard to both forge a new identity and find some independence from you -- so taking steps like dying hair advances both those goals. In fact, the more you fight against your teens wanting to dye their hair, the more attractive it will likely become to them. Regardless of whether hair dying concerns you, you should ask your teen why he/she is interested in changing his/her hair color. Is it to model him/herself after a favorite celebrity or because his/her friends are doing so? You can take this opportunity to talk with your teen about his/her identity and self-confidence.
You should also ask your teen additional questions that will help him/her really think through his/her decision. What color has he/she selected? What will he/she do if he/she doesn't like how his/her hair looks? What does he/she know about the safety of using chemicals to dye his/her hair? Who will be doing it? What changes does he/she expect in how people view him/her or how his/her life will be?
These questions shouldn't be asked in an aggressive or judgmental manner. Simply be inquisitive and respectful; being genuinely interested in how your teen has made such a decision and helping him/her walk through its ramifications will go a long way with your teen. So even if you don't like the idea of hair dying, your teen might be more open to hearing your opinion on the issue. However, even if you are against it, you might want to think about selecting this issue as a battleground. At the end of the day, hair color is a small issue and even if it's truly awful, your teen's original hair will grow back. Teens like to experiment and changing hair color is really one of the more innocuous options.