Will law and order trump sue the NYT for leaking his tax returns to the public or will he just cry on twitter?

Unlikely.

First, the Times was NOT the leaker. They merely reported it. It's unclear who it was.

If it was someone from the IRS or the NY state taxing authority, then law  was broken - but, again, not by the NYT.

Second, the word "sue" implies civil action, which would not apply to breaking of law.

If you meant Sue For Libel there are 2 reasons that is also unlikely.

First, Trump, as a very public political figure is fair game for journalists (Free Speech, dontcha know).

Also, since it's likely true, a rigorous defense against any defamation is Truth.

Any suit brought would be for publicity sake & legally frivolous & probably counseled against by Trump lawyers.

You mean the tax returns from the story FOUR YEARS AGO? I think he's over it. 

Trump is too busy to worry about a liberal, lying rag.  He has accepted this is part of fame and the position as POTUS.  Otherwise that azz wipe of a newspaper would not of printed lies for fear of law.