On a Checkbox formula field, you do not need to explicitly set its value to True within the formula - all you need is for the expression itself to evaluate to True.
For example, instead of using the
IF statement, just start with
Case_Owner_Profile__c <> 'Field Service'.
When that simple statement evaluates to True, the checkbox should be shown as checked.
Test it with a Field Service Profile and a Profile that is not Field Service.
A good place to start is with only one of the Profiles, and work through the logic and issues to get it to work. Then add the other Profiles....
State what you are trying to accomplish in plain language, and look for the logic words: "This checkbox should be checked when the Case Owner's Profile is not Field Service or not Technical Support or not Customer Service."
You can then work through adding to your expresion using the
Ultimately you can try:
Case_Owner_Profile__c = 'Field Service',
Case_Owner_Profile__c = 'Technical Support',
Case_Owner_Profile__c = 'Customer Service'
You don't have to format it that way with the indentation and the operators on different lines; but especially at first when working through more complex logic, it can be much easier to read and follow.