Yes, it does accept my license each time and I am able to edit TaskPaper documents like normal.
I believe I ran the preview/pre-release for version 3, which I purchased in November of 2015. I also tried the 3.7 preview to see if this would resolve my error. I believe the 3.7 preview is currently the version I have installed.
One additional piece of info: I tried to remove any support files associated with TaskPaper but I couldn’t find any in ~/Library or /Library.