I (a grad student) first-authored a paper with my ex-PI and some colleagues. The paper was rejected from a journal. Meanwhile, this ex-PI also bullied me, did something else that was hurtful, and did not provide helpful feedback on the paper. The ex-PI is now asking if I want to re-submit. Can I politely refuse? If so, how?
They’re an ex-PI and one that you’ve had negative experiences with. Still, networking and connections, even if faint, can come back and be helpful directly or indirectly. Diplomacy is key. Give your appreciation for their interest in collaborating, but decline, citing other obligations, time commitments, etc.
If you were happy for the work to be resubmitted more or less as it is but didn't want to spend time/energy doing so, you could ask one of your colleagues to be corresponding author (citing other obligations as in cralr's answer). If you would need to make substantial changes to the work to have it published (if reviewers' comments from the rejecting journal were quite extensive, for instance) that might be more complicated, but another author might be willing to coordinate these changes.
Either way, you'd be well within your rights to say that unfortunately you don't have the time to do anything with this paper. No reasonable person would take offence to this, but that's predicated on the ex-PI being reasonable. An unethical PI might simply take your name off and resubmit elsewhere - you may be able to take this up with the administration of your institution, but it could be a difficult fight.
Sorry you've had what sounds like a really difficult experience.
From your question, it seems that you are not keen on resubmitting the paper. If that is the case, you have every right to refuse. However, make sure that your response is polite but firm. You need not give any reasons or excuses. Just thank him for the request, and say that unfortunately, you are not interested in resubmitting the paper at the moment.
Having said that, however, given your past experience with your ex-PI, I would suggest you reconsider your decision. For all you know, if you refuse to resubmit, your ex-PI might just go ahead and publish the paper without you. While I can understand your feelings, refusing to resubmit would just give him another opportunity to get one over you. If you resubmit the paper, at least you are sure that you will get credit for your work (and an additional publication in your CV). However, make sure that your communication with him/her is polite but firm at all times.
It's extremely difficult to understand your special situation with your adviser. As always, there are at least two sides to every such story, and you're telling us your side of it.
Getting papers through is an important academic goal for PI's, and many uncomfortable situations occur during the process.
I recommend thinking about this in the context of your career aspirations, and not necessarily how pleasant the experience has been so far (though that should certainly factor in). Do you have the time to resubmit, or is your current work demanding all your time? Can you agree on a timetable with the ex-PI?
The way I see your situation from your terse description, it looks like there is an opportunity to recover from an unpleasant situation. You can get a first-authored paper, and maybe patch up a relationship with your ex-PI. If this is important to you, you should consider it, and if it isn't, you should consider telling the PI you're not interested. Since the PI is obviously interested in pursuing this, and you've already submitted once, you might tell them you'd be happy to cede first authorship if they want to take it to the finish line, and you will review all manuscripts promptly, but you have no time for major effort.