One particularly famous example is the Cobra effect. From Wikipedia:
The government therefore offered a bounty for every dead cobra. Initially this was a successful strategy as large numbers of snakes were killed for the reward. Eventually, however, enterprising persons began to breed cobras for the income. When the government became aware of this, the reward program was scrapped, causing the cobra breeders to set the now-worthless snakes free. As a result, the wild cobra population further increased. The apparent solution for the problem made the situation even worse.