What to do when phd advisor wants me to work on side projects but i'm starting a post-doc? I'm currently finishing up writing my PhD while starting a post-doc at another university in another country. The post doc is a great opportunity at a top tier university. Between finishing my dissertation, trying to get started in this new lab, and moving to an apartment in another country that requires I start from scratch, my time is super limited and everything is hectic.

I made sure my post-doc advisor was aware that I'm finishing my dissertation , and would be doing so for the beginning of the post-doc, from the get-go as I informed him during interviews for the position and when I accepted the job. He has been fine with this and held the position for me for as long as he could because as he said I'm a perfect fit for the position and he really wanted me to be his post-doc, so I'm super grateful for that. Plus, there is enough crossover in the project that my PhD advisor made him a collaborator on the chapter that I'm finishing up while starting the post-doc position.

My PhD advisor recently reached out to me about a side project I had been working on for him previously and had written up with me as the first author. This paper was rejected from the previous journal we submitted to and we received this notice around the time I was getting ready to move. Because of this, he offered to re-work the manuscript results/discussion sections after I re-did the analysis and sent him the new results, which I did. Now, he is saying that he doesn't have time anymore and that I need to set aside my work to make this a priority.

The other relevant background information is that my PhD advisor is in trouble with the university because he is so absent when it comes to his students. He had another PhD student who started at the same time as me defend her dissertation before she was ready, and the defense was so bad that department members outside the committee were asking how she was allowed to defend and it got to the department head. Now he is in trouble of losing teaching assistantships for future students. I have been independent for my PhD and have worked on many side projects for him. Just 6 months ago, again while I was trying to finish up my thesis because of the post-doc position, a similar thing occurred where I was asked to set aside everything to devote my time to a project I was a co-author on and it ended up being 2 months of re-doing the entire analysis, results, and discussion and with me being a co-first author on the paper, which was published after my rewrite. I have also worked on numerous other projects for him over the years. I have been very independent in my PhD, not relying on him for much as I came up with the question, methods, and interpretation of my PhD project - my independence (a byproduct of his absence) yet success with this project is a part of the reason why I got an amazing post-doc position.

I think realistically this re-write would be ~1-2 weeks and I know that I was first author on the manuscript and the major person behind the project, but I just don't have the time. I proposed to have a student from our collaborating lab that also did a lot of the work on it just finish the re-working of the results/discussion and to make him co-first author, but my PhD advisor wouldn't entertain the idea. I think he needs more papers coming from his students alone. 

What do I do? Should I bring this up with my new post-doc advisor? If I do this I either won't have time to work on any new stuff with him or I won't be able to work on my dissertation and will fall behind there. Should I say no to my PhD advisor, and if so, how?
I am currently in a similar position to the one you describe, except that in my case I agreed to take on a brand new side project with my previous supervisor. I spoke to my postdoc supervisor about the project as soon as I started and received his go-ahead to spend some of my time working on it. In my experience, it seemed like my postdoc supervisor thought this was totally normal and expected after a PhD.

In your position, I would agree to the rewrite (after verifying with the postdoc supervisor that this is not an issue) because it seems like it will not be too significant a time sink - you already know the work and have a reasonable estimate of how long it will take. It is always nice to publish your work. Note that you do not need to spend the first two weeks of your postdoc on it - you can allocate some time each day to it and then you will still have time to do new stuff.

I have one caveat based on my own experience which is that even if you have permission to spend some time finishing up old work, it is quite difficult to commit to doing this because it feels like you are not putting 100% into the new project. It can also be tiring to have a task that needs extra work after hours when you are trying to get established in a new place. Thus, I would limit the number of "side projects" you agree to bring along from your PhD and try to make a realistic plan for how you will finish them quickly.