The bacteria on the skin can get under the skin and cause a minor or severe infection. An infected eyebrow piercing presents a greater chance of spreading to important structures (the eye, the bone) and causing harm. There have been many reports of infections as a complication of body piercing over the past few years.
The way in which you obtain the piercing may have possible side effects as well. If the instruments aren't sterile and have been used for other people, then it is possible to transmit infections such as hepatitis and HIV when piercing any part of the body.
In some people, any wound in the skin can heal with large, unsightly scars, called keloids. In people who are predisposed to this, a keloid can form right at the site of the piercing, and is difficult to cover up or remove.
Another disadvantage comes later, when wearing jewelry through the pierced eyebrow. Unlike the ear, where people often wear stud-type earrings, most often a ring or hoop is used in pierced eyebrows. Accidents can certainly happen more readily, with the ring catching onto someone or something during regular activities or sports. A ring that is accidentally ripped out of an eyebrow has the potential to cause a lot more damage than one ripped out of an earlobe.
All in all, there are many disadvantages to piercing an eyebrow, or any other part of the body. One should really think hard about whether getting an eyebrow pierced is worth these risks. My own personal feeling is that it isn't.