When you deploy a private Burp Collaborator Server you tell it which domain it controls so that it can serve the appropriate DNS records. This is configured in the Collaborator configuration file. This means it would only be responsible for responding to DNS requests for the collaborator domain. It would not be responsible for any other DNS requests.
So if your collaborator domain is:
collaborator.mydomain.net
DNS queries such as
foo.collaborator.mydomain.net
would need to be directed to your private collaborator server
This link may help your network admin to understand the role of the Collaborator Server and how it works in a bit more detail:
https://portswigger.net/burp/documentation/collaborator#how-burp-collaborator-worksI hope this helps to explain things. Please let us know if you have any questions.