It will not damage your battery.
My guess is that what you are feeling is what's called "touch current." Many electrical devices (switch-mode power supplies especially, which your power brick (charger) is), particularly ungrounded ones, employ noise filtering or shielding capacitors that are connected directly from the chassis ground to the AC neutral line... which, with an ungrounded and unpolarized power cord, or with a miswired AC outlet, can become the line to ground. And the "ground" in the PSU does get connected to "ground" in the laptop. Which is apparently connected to your aluminum wrist rest, probably again for shielding purposes.
A small amount of reactive AC current will flow through such capacitors and if the "ground" isn't actually grounded, some people can feel it. It often seems like a "fuzzy" sensation when you move a finger over the metal. Some people can't feel it - they have a higher threshold for perceiving electrical current.
"Touch current" is required to be under certain limits by safety regs. These limits are generally above the threshold of sensation (hence your being able to feel it) but well below the level that would pose a shock hazard. As equipment ages, the "touch current" sometimes gets worse. But there is almost always so little available current that if you measure the voltage with a high-impedance digital voltmeter you'll get one reading, often half the line voltage, but try it with a lower-impedance analog DMM and you'll get a much lower reading. This is indicative of a very high impedance path for the "touch current", which means it *probably* is not a safety issue.
Still, it's not the most pleasant thing.
One simple fix: If your power plug is nonpolarized, just reverse it in the power outlet. See, the usual offender here is that the shield capacitor, which is supposed to go from chassis ground to AC neutral, is going to AC line (hot) instead. But if you can reverse your plug in the AC outlet that easily, there's no guarantee it will always be plugged in the right way.
A sure fix is to ground your power supply. If its AC power inlet connector allows for a grounding pin (i.e. if it has three or more prongs), buy an AC cord for it that includes a ground prong. If you can't do that, buy a compatible power supply that DOES have a grounded AC cord.