I do not have the new Apple TV or a 2006 Mac mini running Snow Leopard so I have no first hand experience, but I would bet on no for 2 reasons.
1. The major problem is tvOS. Apple TV and the new remote interface via the new Apple tvOS. Macs do not run tvOS. Same problem with Siri, no tvOS. Older Apple TVs also will not have the new tvOS.
2. The new remote connects via Bluetooth. Your 2006 Mac mini has Bluetooth 2.0 and the newer technology lately have been operating using Bluetooth 4.0 so there may be a problem there.
What did you want to do with the new Apple TV remote?