Can a DualShock 3 controller overcharge if it's always plugged into a PC?

I have my DualShock controller plugged into my PC right now, but I'm wondering if any harm will come from just leaving it plugged in at all times. Will it overcharge? Will the battery life diminsh faster? Will the controller run out of power quicker when I play wirelessly?

Do not worry. Dualshock is equipped with Li-ION or Li-POL battery. Lithium batteries are very dangerous and can explode if used improperly (you can go deeper in the topic if you feel interested). So like I said - do not worry, because of exposion hazard the electronics responsible for battery maintenance (charging, discharging) must be secured at least by delta algorithm, current and voltage control. There should be also temperature control. All those security mechanisms prevent you from any harm, and if something goes wrong the battery will be immediately cutted off.

In case of lithium batteries overcharging (or charging with too high voltage) createsthe same level ofdanger as shortcutting or piercing - the battery can (and will) ignite or explode. So the electronics that secure overall usage process of battery is multiple times tested before Dualshock (or any other device equipped with lithium batt) even goes out of the factory :)

And here are some videos of modelingbattery packs getting fire:

• video 1
• video 2

I forgot to tell you - if your Dualshock charges full it will automatically turn off the charging process, no matter what USB host were you using (PS3, laptop, USB charger). And most of common usage lithium batteries have built-in MOSFET transistor protection (just in case of electronics failure). This transistor will get dammaged (preventing further damage to whole device) if too high current is taken from the battery, or the battery is shortcutted.

Sorry for the lecture, but I tought you'd like to know everything

I wasn't aware the DS3 will turn off automatically when fully charged. I usually see the solid red light on the controller lit ON the whole time. Would that indicate the charging is complete?

I assume that at end of charging process the battery is disconnected (in fact it is sustained with very low sustain current), and the LED indicator is powered from USB.

Battery charging is a very broad subject. And as far as my knowledge reaches in this toppic all charging devices are autonomical and very complicated (including DS3). Thing that can negatively affect battery's lifespan are:

• deep discharge (under 3 - 3,2V per cell. With some batteries the low voltage state is even at 3,4V) - DS3 electronics whach over this parameter, so dont worry,
  
• long time storage on fully discharged - the battey will discharge on its own over time. When the battery discharges too deep, it will be unable to charge again (see above) - this parameter isn't monitored, so remember about it,
• temperature of cell. The higher the tempetarure goes, the faster the battery will die. Temperature can rise on it's own from too high current given or taken from the battery - can happen during charge process, but most probably DS3 electronics whatch over it. But don't place your DS3 near a heat source - this can be dangerous.
• Lithium batteries begin to get older after 300 cycles (NiMH batteries begin to get older after 1000 cycles), after that each subsequentcyclereduces thecurrent efficiencyand capacity.
  

Overcharging, like I wrote earlier is absolutely prohibited, in licensed devices like phones, music players, or Dualshock and won't happen.

In other words Dualshock has a built in autonomical charger and safety features that will monitor the overall usage process.