Am I obligated to finish my post-doc term before starting my tenure-track job?
Congrats on getting a tenure track position! 

You are not the first person to whom this has happened.  Of course, there may be specific circumstances that make the early departure more difficult - say, if you are the only person who can operate a certain instrument in a lab.  But people do leave post-docs early, from time to time - it's not at all unheard of.  And, if you applied opportunistically, you probably have a job that you are happy to take. 

My advice is to break the news to your advisor as soon as possible - once you have made the decision to accept the offer. Your current institution might still be able to find a replacement for next year, which would be good for them. If you reflect on it, it's hard to see real advantages of telling your advisor later.  And there is a good chance they will be happy for you that you got a tenure track position. 

Remember that you will still be in touch your advisor professionally, so you want to try to keep the relationship on the best terms that you can.