Fired from TA position without cause/notice, what to do next?

Even my PI in research wanted to contact them because he thinks I am not treated fairly. What to do next?

First, you should relax. Being a TA is not a big deal.

Second, you should wait. This is a situation where your PI ought to be responsible for helping you out. They already said they would try, so let them try.

If your PI does not succeed in helping you, then you might follow the university's official complaint procedure. Keep records of everything just in case.

From what you say, it does seem like you did nothing wrong and were nonetheless fired. I would call this an "injustice", but I can't see where anything bad happened to you.

• Your post-doc contract was not violated.
• You were not financially impacted. On the contrary, it sounds like you get the money without having to teach, which is a win.
• You mention some concerns about your resume, but this is frankly nonsense -- no one is forcing you to write "I was a TA but got fired" on your resume. In reality, no one will ever know.
• You perhaps have some legitimate concerns about your reputation. Sitting down with the department chair or someone responsible for the department's teaching may be helpful. You may also want to ascertain whether there is anything in your file that could adversely impact your reputation later (e.g., anything that would show up if you applied for a reference -- or even a security clearance).
• Are you eligible to teach in future semesters? If not, then you perhaps have grounds to file a complaint and try to get your eligibility reinstated. But you would have to follow the university's procedures -- I doubt you would have any legal recourse since you weren't fired due to race, gender, etc.

It is good that your research PI is willing to help you. If you have a specific "damage" you would like to be compensated for, I would ask for their assistance in achieving this. But "seeking justice" is likely not actionable -- instructors are usually allowed to handle their TA assignments however they want.