Is it ok to share a a client's dataset on github, along with my code?
My take, as someone who tends to use other people's data in extended analysis, and does enjoy having the most up-to-date version of an analysis/etc. up on GitHub:


  Should I ask each user if they want me to remove the datasets from
  their repository?


Yes. You should, fundamentally, be respectful of other people's data. Which includes things like "I'm about to put this on a website". While you haven't mentioned if your repositories are public or private, there's no guarantee that data stored in a private repo won't end up in the wild, one way or another.


  Should I remove all such datasets regardless? Why or Why not?


I don't know that I'd remove all the data sets that are already up there, but I would definitely ask for approval, and if you don't hear back, assume that the answer is "No".


  Is there something else I might be missing about this scenario?


Yep...


  These are mostly medical and health research projects.


You haven't said what kind of data, but beyond just whether or not your anonymization step is actually sufficient, is whether or not the agreements that generated this data - IRB approvals, informed consent documents, data use agreements, etc. actually allow it to be put online in this form. That is, by no means, a sure thing.