I know how I can assign a test case to a specific user. Is there a way to assign it to the role "tester", so that any free tester can execute it?
I haven't used TestLink a lot but, as per what I have used it seems not feasible. Because you can assign test cases only to a Role-User (a person generally from Tester role) for execution and update, but it can't be assigned only to a Role.
For resolving your issue and for achieving what you are trying to do, you can create a dummy user (Passive user) which should not be existing in realty but only in your system like 'Community user' of SQA and assign it Tester role. So, any test case(s) assigned to this user should be considered as 'Open for execution by any tester' test cases and any tester can pick them for execution.
If you see that there is no such feasibility in your project to create such a user then you should assign such test cases to either QA Lead or Team Lead (one who is not responsible for execution) and treat those cases as 'Open for execution'. This will not require any new user or anything else, such you need to inform all for this change of process.