Seems like the tape would get burnt up before too much damage was done though.

Yes, but across sufficiently high voltage lines the initial arc would create a plasma channel which would act like a thick wire for the short to continue

I guess it's probably not enough to sustain a long time and there are probably protections in place to prevent that.

Here is a short video showing someone doing almost exactly this: https://m.youtube.com/watch?v=X_2lZqyCmls

Pretty confident that video is fake. The concept holds though. This video seems more realistic. https://youtu.be/2NKOlLDNknc

Point being the initial path creates a plasma channel that then bridges the gap and lets current flow.

So I would expect an outcome like 3 or 2 in this video (where something bridges the lines, established the arc, and then it goes away):

https://www.youtube.com/watch?v=gOT8jx4jEzM

Is the arc dissipating because an automatic system is killing power to the lines? Is there going to be damage elsewhere in the system from events like 3 and 2 in the video there?

No it works pretty well. You're essentially making a thyratron.

https://en.wikipedia.org/wiki/Thyratron