What's this? What is it? but not What's it? - Why?

It's a perfectly grammatical contraction, though it doesn't commonly stand alone; it's usually followed by another word.

• What's it all about, Alfie?
• What's It? - The Award Winning Game Where Creative Minds Think Alike!
• Sun, sea and silver service: what’s it like crewing on a superyacht?
• What's It? - Information Today
• What's It Like on the Pope's Plane?
• What's it to you, anyway?
• *What's it do?"

The problem with your example is that in the short, stand alone phrase "What is it?" (just as with Who is it?), the emphasis is on is, not what. If there is no emphasis on is, then the phrase is simply What? If the emphasis is on it, then something for the dummy-it must be stated, as in the cases above, or, What's it like outside? (Who's it gonna be?)

edited to add: Please see @John Lawler's comment.

What is it?
What's this?
What's that?

Why are people so averse to providing context? English lives and breathes through context.

The proper way to approach this is to ask how the expressions are used, not to examine them as isolated phrases that can be poked at and dissected without any reference to real-life usage.

Example

(a) Jane points at an object lying on the table. She says, "What's that?"

The word 'that' is strongly stressed. We understand that something new has come to Jane's attention. The something is that. The verb 'is' cannot take a position of stress because that is already occupied.

(b) Jane points at an object lying on the table. She says

"What is that?"

This time, the word 'is' is strongly stressed. Jane is talking about something that is already under scrutiny by those present but now she is saying, "I see it but now I want to know what it is" Clearly we can't abbreviate 'is' in this case.

Now all we have to do is apply the same reasoning to 'it'.

(c) Someone hands an object to Jane, she enquires, "What is it?" This is similar to case (b). There is a known object and Jane wants some information about it.

(d) Now we come to the 'impossible' case. There is an object.

Jane says, "What's it?"

This should be like case (a). That is to say, the object is new to Jane. However a vital property of the word 'it' is that it must have an antecedent. Therefore Jane must already have mentioned it. So it can't be like case (a). This case doesn't happen because there can never be a context that warrants it.

Conclusion

You can only understand the reason by considering the contexts in which the expressions can be used. You cannot do it by simply reasoning about the isolated phrase—English requires context.

Unlike for 'this' and 'that', there is simply no context that allows emphasis on the word 'it'.