Should I turn off my PC when plug/unplug external display?

Short Answer
Assuming that this external display input is VGA, DVI, or HDMI there should be no adverse effects.

Rationalization / Support

• All of these connections are data only, so there shouldn't be any large captive or inductive loads built up on the lines.
• I have never went out of my way to shut down my PC to plug or unplug a monitor / projector in 12+ years.
• A bit more support of this would be using a KVM you remove the signal to the displays as part of normal operation. Granted that it stays plugged in at all times (see warnings below regarding the physical motion of plugging the device in).

Areas of possible concern

• If the external display carries power to the device (meaning you do not have to plug it into an AC outlet)
• In the case of a PC tower and a PCI type video card. If the screws are not installed or not tight you run the risk of inadvertently unplugging the card from the databus which could cause the system to behave unexpectedly. (in Windows can cause the blue screen of death)
• aland's comment on 3 phase power brings up a good point. There can be situations when the display and pc are connected to different outlets that could cause serious issues. In short if the notion of "ground" is not the same for the display and pc it could damage both of the display and laptop. While this damage would occur even if you powered down your display, it's still worth noting. I have seen this very issue with a cheap China direct Dell power supply blow up USB devices because it's notion of ground was 60V below the device's ground reference.

Closing Thoughts (Almost)
If the external display is a 'normal' display and all of the hardware is installed per the manufactures recommendations you should have zero troubles. You could always look up the video card / laptop's manual as well as the monitor's manual to be safe.

EDIT

Additional Thoughts
After reading sawdust's answer and doing a bit more digging I happen to come across a duplicate question here on SU. The more I read on it it really boils down to a few points.

• Because the nature of electricity, it will be hard to find a display that states that it is 100% safe to hot plug the device.
• While there may not be a data sheet that shows it, but when so many people (dare I say hundreds of thousands?) have done this for years with no trouble it is hard to dispute a give track record this positive and large.

When trying to answer the question "Should I....", it becomes a calculated risk (as any real world problem is). Take a moment to answer these questions and it should give you your answer. There can be many more variables to the calculation, but this should at least get the point across.

Business Environment Variables
1) How long does it take to power down and up an average business grade laptop?
2) How much does it cost the company per minute of your time?
3) How many times per day would you connect and reconnect the display?
4) What is the cost of a new PC + display?
5) What is the percentage of 'safe' operation?

Do you see where this is going? By powering down and back up each time you want to move your machine several times / day it could quickly become evident that buying a new machine + display would be cheaper in the long run. Please note that if you switch a lot during a meeting with a number of employees, the cost per minute skyrockets.

Personal Variables
This is much more simple...
1) Do you want to minimize all of the possible risks?
2) How patient are you?
3) How much faith do you have in the proven track record of hot plugging the display?

Closing Thoughts
While I can say that there is a not 100% certainty that nothing bad will happen to your display / machine, IMHO the time savings and large proven track record makes the risk a very acceptable one.

Consult the owner's/operator's manual of the display device. Most likely the manual states that both display and PC should be powered off when connecting them. Obviously this would be the safest and no-risk method.

The capability of connecting devices while one is powered on (aka hot-plugging) is partly determined by the connector design. If the electrical ground connection(s) cannot be established before any power and then signal lines are connected then there would be the chance of damage to line receivers. Inspect a USB device connector and you'll see that the contacts are not all of the same length; note that USB devices are hot-pluggable.

Neither VGA (HD-15) or DVI connectors are designed for making ground before signal, so there could be some risk when connecting powered devices. Just because someone has done it without problems does not mean that you never will. BTW I've seen electricians work on live circuits, i.e. they don't shut power off to replace a outlet; just because it can be done does not mean it should be recommended to others.