Would using an iPad as a second monitor be harmful to its battery life?
The battery will wear faster when you use power-hungry applications or set the display very bright very often. Duet may be such an application.

You may not notice it, but the battery transfers much more energy in the same amount of time. This leads to more 'battery cycles' which in turn degrade your battery.

It has no negative impact on the battery life if you keep the Lightning cable in plugged in all the time.

However, use the original 12 Watt USB power adapter and a certified Lightning cable for your safety, especially if you leave the iPad unattended.



General tips to increase your iPad's battery life


Keep the device cool. Remove unneccessary cases and don't place it near warm objects.
Use certified Lightning cables. This is for your safety. You don't need to expensive cables made by Apple, as long as they are certified. They are called MFi which stands for "Made for iPhone/iPod/iPad". MFi cables undergo electrical and mechanical tests by Apple.
Use original power adapters. Again, you don't want the power adapter to catch fire.
It's fine to connect the charging cable to your computer's USB port instead. 
If your iPad will be unused for some days, discharge the battery to ~50 %. "(…) if you store it fully charged for an extended period of time, the battery may lose some capacity, leading to shorter battery life."




Terminology

Battery cycles


  When you use your [device], its battery goes through charge cycles. A charge cycle happens when you use all of the battery’s power—but that doesn’t necessarily mean a single charge.
  
  For example, you could use half of your notebook's charge in one day, and then recharge it fully. If you did the same thing the next day, it would count as one charge cycle, not two.


Source: Apple, emphasis mine

Battery wear/degradation/condition/health

The more you use a battery the more its capacity will reduce over time. This is sometimes called 'battery wear'.

Battery degradation can have different causes. Using a device so often that you need to re-charge it daily from 0 to 100 % will lead to a high battery cycle count. The chemicals in the battery will store less energy after each cycle.

The battery condition can give you a hint how much of its original capacity a battery can hold. It may be ok for a battery to retain about 80 % of its original capacity after 1000 full cycles. If you have a new device with the same low battery condition (80 % capacity) something may be wrong with your battery. This can be caused by faulty battery or a few cells within your battery that are not working.
Battery condition is sometimes called battery health.