Potential consequences of providing voluntary work for a team I've been forced to leave
Personally, I think you should focus that time and effort into getting another position, if you have not got one already.
If you have a new position already, well done, then focus that time and effort into where you are now. They, hopefully, will appreciate that effort and motivation.
I would strongly recommend to ask your former PI, mentioning that many users asked for a video and that you would volunteer to produce it without any strings attached.
If he agrees, go for it. If not, move on.
I would not do it without consent (even though you might have the right to do it as long as you don't mention secrets).
First, sorry that you were being forced to leave, hope you can find a better place to work.
As for your question, I'd like to think more objectively by asking these questions:
- Do you feel you will enjoy creating that tutorial?
- Will that tutorial be useful?
- Do you have time for creating the tutorial?
If the answer to those questions are "yes", then I guess it would be good for you to create the video tutorial, as it will also help users in fully utilizing the software, which will help their work as well.