Sometimes it can be enough to just clean the contacts without having to replace the entire board. Dosent work for complicated IC's like plcc type, where corrosion is underneath the chip. Here you will have to reheat and reapply the IC.
Yes, damage can occur. Always disconnect any kind of power before removing different boards. After removing battery connector, it can be a good idea to discharge capacitors by pressing the power button on your mac. This ensures theres no latent current passing through the circuits.