This works just fine on AEM 6.5.6 for me.
Try issuing these Curl commands (updating user/pass, path as needed)
- curl --user admin:admin http://localhost:4502/api/assets/test.jpeg -X PUT -H"Content-Type: application/json" -d '{"class":"asset", "properties":{"dc:title":"My JPEG"}}'
- curl --user admin:admin http://localhost:4502/api/assets/test.tiff -X PUT -H"Content-Type: application/json" -d '{"class":"asset", "properties":{"dc:title":"My TIFF"}}'
- curl --user admin:admin http://localhost:4502/api/assets/test.tif -X PUT -H"Content-Type: application/json" -d '{"class":"asset", "properties":{"dc:title":"My TIF"}}'

If you are able to download the original rendition and open it (showing there is nothing wrong with streaming the bits off the original rendition JCR node) then im not sure ... Since you can update other Tiff's it sounds like there is something weird w/ this Tif's manifestation in AEM. You could try re-uploading the downloaded orginal as a NEW asset and see if that asset has the same problem. If so, maybe there is something w/ that tiff? All and all, HTTP Assets API supports PUT requests for Tif's. If you need to understand why this specific Tif is misbehaving, Adobe Support should be able to help you dig deeper.