You need to set the audio out to Bluetooth. 

https://usermanuals.finalemusic.com/Finale2014Mac/Content/Finale/PRECCONFG.htm

A few issues with the airpods.

First, it is frustrating that they do not immediately/automatically connect. I can't just put them in my ears, and Finale knows to send the audio to them. I have to do something manually, like setting the audio out to bluetooth.

Also, after using for a while and I take my airpods out, the audio disconnects and won't reconnect without quitting Finale and restarting it. It gives error code 7.

Error code -7 is a generic MacOS code that indicates a problem. You know how to cause it.

Otherwise, not enough info. You have a number of options for audio output in Finale. You need to be set up so that System Audio is the default and have that default to BT ‘phones whenever connected. There are a number of ways to do this with control panels and preferences. Google is your friend. If you detail your setup exactly, I may be able to see the problem.

I've been having this same problem for years. I have no problem hearing Finale through my AirPods, but when I take the AirPods out and put them in the case, macOS removes the AirPods from the current audio configuration, which is by design. However, Finale can't handle the change in configuration, and throws an Audio Engine error.

Because of this, if I need to take my AirPods out or even just walk out of Bluetooth range, I have to save my work and quit Finale, and then reopen and reorient when I sit back down. Worse yet, if I forget to do so and leave Finale running overnight, I have to spend the first few minutes of my morning dismissing the dozens or even hundreds of Error -7 dialog boxes that have piled up!

Mike suggested to set Finale's audio output to "System Audio," but I don't believe that's an option: under MIDI/Audio > Device Setup > Audio Setup..., the Source menu under Output lists only the currently active channels. Perhaps there is some way to manage this using a utility like Loopback or something but it would be nice to just have Finale handle the configuration change more elegantly.

And Google usually is my friend... but not in this case! I've experienced this problem consistently since at least Finale 25 and have never found any solutions despite years of looking.

Toby is correct - I'm on a Mac and I can't see any way to set the output to System Audio or Bluetooth. 

Every time I take my air pods out or move out of range I get an error code 7. 

Shut the lid of my laptop - error code 7. 

This means every time I take a break from working on something, I have to save and quite Finale. Really annoying. 

Anyone got a workaround for this?

I have the same issue, but it also includes my iPhone as well as AirPods. If I do something to disconnect the AirPods or iPhone from Finale's "microphone", Finale throws that Error -7 dialog and there is no getting past it. Today, I was listening to a webinar in Safari, using AirPods, and working in Finale while I listened (copying notes from a printed score to a part). When the webinar ended, I put away my AirPods, and lost all the work I had done in Finale to that point. I might dig around and see if I can find an auto-saved document, but still, this is annoying. Finale should just notify us that the audio connection has changed, and not lock up like it does now. Using the newest version of Finale, v. 27x

Again, -7 is a generic macOS error that goes back to the beginning -- long before wifi, bluetooth or even networking.

It normally means that something (an app or a system process such as audio or bluetooth) can't find something else (a folder, directory, file, or in this case, an audio playback or input device). Finale might freak out but it is the macOS that is throwing up the error that Finale can't find what it needs. The only "fix" is not to disconnect or or shut down the process or move the external device out of range. 

Bluetooth has a limited range on purpose.

I get all that, but how do Logic or Final Cut handle it (I get that these are Apple products, but I’m thinking if it’s no problem in that software, why can’t Finale handle it?)

but how do Logic or Final Cut handle it (I get that these are Apple products, but I’m thinking if it’s no problem in that software, why can’t Finale handle it?

MakeMusic would have to answer that but I suspect that you already have. Many non-Apple apps have this issue. MM never explains such things on these boards.

I'm only the messenger and neither work for MM nor Apple.