I don't think you can remove the push URL, you can only override it to be something other than the pull URL. So I think the closest you'll get is something like this:
$ git remote set-url --push origin no-pushing
$ git push
fatal: 'no-pushing' does not appear to be a git repository
fatal: The remote end hung up unexpectedly
You are setting the push URL to no-pushing
, which, as long as you don't have a folder of the same name in your working directory, git will not be able to locate. You are essentially forcing git to use a location that does not exist.
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