The basic theme of advice is to tin the tip of your iron first. It helps it quickly conduct the heat to your wire you wish to free if you have a little dab of liquid solder on it. Simply give you a larger surface area of hot contact.
If you keep putting a hot iron on the board and not getting a quick melt on the solder you will cook something sooner or later.
My advice if this keeps being an issue is to go to a hobby shop and consult somebody there in person. Good chance they can offer help beyond what you can read online. Other than that, hit up youtube with searches for soldering wires to a circuit board.