This can be done, although there is the possibility of doing further damage to your console.
Soldering irons are generally not used for repairs such as these, the joints are too small to use standard soldering techniques.
There are two basic techniques for repairing problems with PS3 solder, reflowing and reballing.
Reflowing is a process where the board is heated up and the original solder is allowed to soften. This allows the breaks in the solder to reconnect. When the heat is removed, the solder will cool down and solidify, and if all goes well, the PS3 should work. I have friends who have successfully done this, but I have never done it personally.
Reballing is when the original solder is removed from the mainboard, and new solder is applied. Apparently, this procedure requires special equipment, and I don't know any amateurs who have done it successfully.
If you want to try your hand at fixing your PS3 yourself, it seems like reflowing is what you want. Google might have some tutorials on how to do this, but I can't give you any advice on what specifically to do. Remember that opening your PS3 voids the warranty, and that when you mix heat with electronic equipment, you're as likely to break it as you are to fix it.
Also, keep in mind that electronics can fail for reasons other than broken solder. If your PSU has gone bad, for example, reflowing will do nothing at best, and be harmful at worst.