Green tea has been heavily over-hyped. Even mainstream tea companies, including Teavana, make a lot of unsubstantiated or exaggerated claims about the health benefits of their teas, especially green and white teas.
For accurate information about tea and health, I recommend reading the page on tea on the micronutrient information center of the Linus Pauling Institute. I trust this source, an impartial resource institute, much more than companies which have an interest in promoting tea.
The strongest evidence supports that consuming at least three cups of tea daily may modestly lower your heart disease risk. Although there is a lot of hype about tea preventing cancer, there's not enough evidence to conclude that it actually prevents cancer or lowers cancer risk. The same is true of weight loss.
Health risks or drawbacks:
Both green and black tea contain caffeine. Although the amount is quite low, much lower than in coffee, the amount of caffeine in tea can still be enough that it can cause problems for caffeine-sensitive individuals, and for some people, it can contribute to insomnia if you drink tea too close to bedtime.
Both types of tea can be high in fluoride among natural food sources. Andrew Weil took a stance on this issue, explaining that, while green tea does have about twice as much fluoride as black, he thinks concerns about fluoride in tea are overblown. The fluoride may contribute to positive dental health, however!
There is also a little concern that tea may inhibit iron absorption, due to the tannins or other antioxidants binding to iron. There's not much evidence that this actually causes a problem in normal, healthy individuals. I researched this some time back and summarized my work on RateTea's page on tea and iron absorption. If you are really worried about this but want to drink tea, you can take measures like drinking tea between meals, and making sure your iron-rich meals also contain natural sources of Vitamin C.