My Zotero did not update, showing a dialog that while there is an update available, I apparently lacked the permissions to update. First I tried to chmod the directory where I extracted Zotero to, and the Zotero binary, but that did not change the error message. It turns out there is another directory that needs to be changed:
sudo chmod 707 /usr/bin/zotero && sudo chmod -R 707 /opt/zotero
(in the terminal) and now everything works as it should.
Update 10 April 2019: see comments for a more appropriate handling of the permissions.