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The 501 error has nothing to do with Heroku.
I believe it to be the setup of your Resource.
There is a bit of a gotcha with TastyPie, and I would refer to the following line from the documentation:
“For PATCH to work, you must have put in your detail_allowed_methods setting.”
i.e.
detail_allowed_methods = [‘get’, ‘post’, ‘put’, ‘delete’, ‘patch’]
Source:stackexchange.com