If I install an aftermarket "Always on Display" app, will it cause excessive battery drain?

Yes but also no.

Basically, AoD is purely software. It does not require special hardware (you can even use it on an LCD screen, although it will use significantly more power and does not make much sense). This is why there are plenty apps available that let you keep your screen on.

At the same time I would also answer the question with no, because even though it is purely software, it's not just an "app". If you use one of those AoD apps, they will keep your device awake. All the time. This will use a lot more power compared to a device with AoD out of the box, which still allows the device to go to sleep while keeping the screen on, which saves a lot of energy.

However, depending on the device and how aggressive the power management is, you might not notice much of a difference. On my OnePlus 5, with fairly aggressive energy saving options, an aftermarket AoD app does not use a whole lot of power (about 1% to 2% of the daily power draw - comparable to a couple of minutes of surfing with Google Chrome).