Not sure I'll be able to clearly convey this, but hear me out.
「確か、明日は先生の誕生日ですよね。」in and of itself is grammatically correct. If you were to say this to your teacher, nobody around you would look at you funny.
In your example, that「でした」denotes a fact you had assumed to be true in the past. You are saying that "you remember, at a certain point in the past, that your birthday was this particular day."
As one of the commenter pointed out, the difference here would be translated into English as:
「ですよね。」"I believe tomorrow is your birthday, right?"
「でしたよね。」"I believe tomorrow was your birthday, right?"
Nothing really special here if you speak English. it's pretty much exactly the same interpretation in Japanese.
You could also look at it from a confidence stand point. You are much more confident when using「ですよね。」than「でしたよね。」 - again, the difference would be something like "I believe tomorrow is your birthday" compared to "I believe tomorrow was your birthday."