Should I tell my supervisor that I added his/her name as a reference for my next academic position?

You should ask anyone who you intend to serve as your reference if they are OK with it, before doing so. It’s not an academic thing specifically; the same would apply in other situations.

As gnometorule’s answer says, in general you should ask permission in advance (for a reference, and generally for anything which is implicitly making a commitment for someone).

But the question asks about a situation where you have already given someone’s name as a reference without asking them — partly through inexperience, and partly because of being put on the spot after the interview rather than asked earlier in the application. Given that, I suggest writing to the referees as soon as possible to let them know, apologise for not asking first, and check whether they’re OK with it. The letter could look something like the following:

Dear XXXX, Following an interview for YYYY, I was asked to provide some names of possible references, and I gave your name as one possibility. I’m sorry I hadn’t asked you about this in advance, so I’d like to check now whether you are happy to act as a reference for me? If not, please let me know, and I can contact YYYY to take your name of the list.

The given situation isn’t ideal, but it’s not terrible either — it’s quite understandable how it arose, and I don’t think most referees would be particularly upset by it. Just try to avoid it happening again.