If you have the Require a pull request before merging branch protection rule enabled on the source branch of a pull request, you cannot resolve merge conflicts via the web interface. Unfortunately, no error alerts you to the failure.
The only solution I’ve found that allows you to resolve merge conflicts between branches with branch protection rules without disabling the branch protection rules is to create a temporary branch, locally resolve the conflicts, and then merge that into the target branch via a PR.
Dan Cook 
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