The meaning of the sentence is this: I saw something in the museum, and I will never forget that thing.
If you use "would", a past verb, it means that "never forget" only happens in the past. That means, you may have already forgotten the thing in the museum. This doesn't make sense because of the word "never", which extends into the future. The intent here is "until the end of my life", and obviously if the person is still talking, their life isn't over.
The only time it would be natural to use "would" there is if it's someone who is near death looking back on their life, noticing that after seeing that item in the museum, they would never forget it.