Elizabeth,
You can use both here. It depends on what you are trying to say.
It was a wonderful day that I would never forget.
( Here the speaker is talking about the past. It is a look at the future starting from a time in the past.)
example
My first birthday party was a wonderful day that I would never forget for the rest of my life.
It was a wonderful day that I would never forget.
(You can also use would when talking about your desire or inclination.)
example
You might forget that it was a wonderful day, but I never would.
It was a wonderful day that I will never forget.
(Here you are talking about an event in the past and making a prediction that you will never forget it in the future.)
example
My wedding last year was a day that I will never forget.