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I think "Tom played Mary a joke." is not natural and "Tom played a joke on Mary." is natural. What is your opinion?

OP is quite right that native speakers normally avoid the "ditransitive" construction (direct and indirect object, no preposition) with play [a joke]. Google Books has just 2 written instances of (She) played him a joke. We much prefer the "monotransitive" version of that, with an explicitly-specified preposition (there are 2220 hits for ...played a joke on him).

But I should point out that Google Books has 5450 hits for the ditransitive (She) played him a trick, which is almost as common as monotransitive ...played a trick on him (5920 hits).

Some native speakers (mainly Americans) will feel a bit uneasy about the ditransitive usage with trick - it really depends on whether they draw a parallel with the non-idiomatic play (him) a trick or the perfectly natural play (him) a song.

To the extent that there's any "logic" involved here - as opposed to a totally meaningless choice based purely on what one hears other people say (a matter of "idiomatically established" usage) - I think the difference turns on the nature of the relationship between the indirect object (him) and the direct object (joke, song).

If we do include a preposition with song, we can choose between to and for. But we can't use on, because that doesn't work with a "Benefactive object" (him, where he gets the "benefit" of hearing the song).

The link in that previous sentence is to a closely-related ELU question, where professor of linguistics John Lawler asserts that the prepositionless ditransitive form is only appropriate in "benefactive" contexts where the indirect object (the "beneficiary") ends up possessing the direct object (which he doesn't with door, joke). So according to him, Please open the door for me cannot validly be transformed into Please open me the door. But if you follow that link, you'll see that not everyone agrees on this point - in particular, BrE speakers (such as myself) are more relaxed about such usages than AmE speakers.

I'm not really an expert on using mathematical expressions in Google NGrams, but I think if you compare this AmE chart and this BrE chart (for relative prevalence of played a trick on him and played him a trick), what it shows is that the ditransitive form is twice as likely in BrE.

TL;DR - The ditransive (prepositionless) form is fine with play + song, but not with play + joke. But things aren't so clear-cut with ditransitive play + trick (which is more likely to be acceptable in BrE than in AmE), so if you want to play safe you should probably think in terms of including the preposition in any context where you're not sure.

EDIT: Reflecting comments made below, I should point out that the specific ditransitive usage play [someone] a trick has massively declined in popularity since the 1800s. So it would usually be seen today as a "dated" usage - another good reason to avoid it if you're not a native speaker who's well aware of such implications.

You are correct. The following is not a natural way of saying that he played a joke on Mary.

Tom played Mary a joke.

As a native speaker, the instant I saw the title I thought, "What equipment did he use to play the joke on? His mobile phone maybe?"

If you play [person] a [thing] it means that you performed something, like a tune or a video.

Examples

Tom played Mary a tune. (He took out a musical instrument and played a tune for Mary)

Tom played Mary a joke. (He went onto Youtube and found someone telling jokes. He played the joke being told on that video for Mary)

I agree with you. In fact when I was reading the question I also thought that Tom played a joke on Mary was more natural even before I read that in your question.

You could naturally say 'Tom told Mary a joke' which, although different in meaning, is constructed the same way, and it is entirely natural.