This was solved by Microsoft support.
The problem was that on our site, we had set the "Title" column as required for all content types inheriting from "Document". The forms for workflow-tasks (.xsn) inherited properties from "Document", hence the "Title" column was required.
But there is no place to edit the Title-column, so it was never filled in. Therefore the xsn-file was always checked out, and user's could not see the edited form.
The solution is
Go to "All files" i SPD
Right click on the Workflows folder
Double click on the "User Workflow Document" content type
Click "Allow modifications" and save in Designer to make that stick if not already checked.
In the ribbon, select "Administration Web Page"
You will see the columns list.
If you have a required column (like Title), Click on the column name
Select No for "Require that this column contains information" and OK
Then select "Optional" under column settings
You might get an error that says you can't change the settings but the changes will take place and the title column will become optional
Delete and recreate the custom form.
Now you can edit the form in designer, and publish changes, and the other users can open the custom forms.