One thing to note about solder wick is that it is expensive, and because it is expendable, a solder sucker may be a better choice if you plan to do a lot of desoldering. I personally prefer to use a sucker to remove most of the solder, then finish up with the wick. Remember that when desoldering, the resin in the solder and the coating on the board may releases fumes. These fumes are harmful to your eyes and lungs. Therefore, always work in a well ventilated area.
Hot solder is also dangerous.
Be sure not to let it splash around because it will burn you almost instantly. Eye protection is also advised. There isn't really too much to worry about when removing solder. Just make sure to get any grease, varnish or glue off the joint before you start heating. If you don't, you will probably foul the tip of your soldering iron pretty quickly.