Is your Mac mini one with an HDD or a Fusion style drive? I am wondering if the vibration is the heat sink vibrating because of the hard drive, since there wouldn't be any other moving components in the device without the fan.
The heat sink should be mounted quite solidly to the board both by the screws around the CPU and another screw at the end of the heat sink. I would see about running it with the bottom panel off just to see if you can isolate where exactly the noise is coming from. Make sure screws are tightened appropriately since a loose screw could be allowing wiggle room for the heat sink.
Alternatively, you could nix the hard drive, if that's what is causing the underlying vibration and replace it with an SSD instead.