The only real danger I could think of is if you have a database administrator working for you and it says as much on LinkedIn, you could then check that DB admins qualifications that they have also provided on LinkedIn and determine the specific tech stack that your company is using. Example: John Smith is an expert in Oracle 11, John works at your company, for an attacker, they would then focus on potentially exploiting that system version.

I guess in the worst case someone could find out you're associated with an unpopular or controversial organization. I think the paranoia about someone finding your org chart is pretty extreme but I've worked at a place where they were like that.