EDIT: Still looking for a complete answer. Please read the edit below.
This is something I've been wondering for some time now. And this is also not related to relative temporal nouns, I understand why you cannot use に with those.
○「2月14日にデートするつもりです。」"I will go on a date on February 14".
○「2月14日『は』忙しいですよ。」"I'm busy on February 14".
○「2月14日に外食します。」"I'm eating out on February 14".
○「2月14日『に』予定がない。」"I have no plans for February 14". Notice how ない could be considered an i-adjective.
○「2月14日に会いましょう。」"Let's meet on February 14".
○「4月23日にもう一度来てね。」"Come again on April 23".
○「11月14日に生まれました。」"I was born on November 14".
But if you wanted to say:
1) Is there a grammatical reason why this is wrong or unnatural? 2) Is this related somehow to i-adjectives? (What about adjectival nouns?)
3) Where else can I expect this behavior?
4) When should I use に to mark specific times? It doesn't seem to work exactly as it does in English for example.
Please, if you have any kind of resource, I will appreciate it. Even those in Japanese are useful to me, thanks.
Here are some threads I read that might be somewhat related:
After thinking about this for a few days, I find that my initial assertion on relating a native English speaker way of thinking with a native Japanese speaker was fundamentally wrong.
This might be more similar to English that I initially thought, for example we cannot say "I will be busy at/in February 14" which would might very well be a better analogy to using に in the Japanese sentence.
However, I'm still looking for an answer to why does using に in a sentence like「2月14日に予定がない。」is allowed? (consider that ない can be both a negative of a verb and an i-adjective). I assume that this is related to an implication in language from reading Mitsutoshi's answer, the implication being that your state from being "busy" to being "non-busy" has changed (an implied instance of changing states), however, this is faulty because the same could be said for the sentence like「2月14日に忙しい。」where the difference between using に vs は would be an expressed "something has changed and I'm busy that day".
I would like to hear more about this from other people.