There are two sides to this coin and depending on which side we're talking about can change the answer completely, the two sides are Tort law, and criminal law.
Another important area to look is it an individual being subpoena, or is it a company own by an individual, these two things don't share the same type of protection against a subpoena
Short answer: There are avenues that both types of laws use to obtain privileged information such as tax returns, but since the subject is the President different rules might apply(e.g. rules for the Executive branch might apply), my answer may not properly address the fact that the subject is the President.
Subpoenas in Civil
For the use of the taxes in a civil case (tort brought by another private citizen), a subpoena duces tecum (A fancy way of saying disclose documentation) can be brought to produce the tax returns or face punishment by the court in the form of fines, or jail time. This can smoothly be done because most civil courts don't see tax returns as privileged information.
In this context would this rule apply to the sitting United States President? Well, the answer is yes, the sitting United States President can be sued civilly for action taken before, during, and after a Presidency, although the President does maintain immunity from actions taken in an "official" context. This rule was adopted in The Supreme Court's 5-4 decision Nixon v. Fitzgerald stating:
the President is entitled to absolute immunity from liability for civil damages based on his official acts. The court emphasized that the President is not immune from criminal charges stemming from his official (or unofficial) acts while in office.
Civil cases against the President
Subpoenas in criminal
Subpoenas in the criminal contexts are alittle bit different due to fifth amendment violations.
Subpoenas for an individual
being Subpoenaed as an individual in a criminal context makes you a witness, and although you have to testify you may take the fifth for questions that might self incriminate.
If you are the subject of the criminal case you aren't subpoenaed to show up, you are arrested and charged, and if the prosecution needs relevant information from you it isn't subpoenaed in is obtained through a warrant.
Subpoena for a company
In a criminal case against a company, companies do not have protections under the fifth amendment, furthermore individuals action to hold documentation for a company also doesn't have protection under the fifth amendment.
Subpoena for the President
There have been three times in history that a sitting President of the United States has been subpoenaed, Thomas Jefferson, Richard Nixon, and Bill Clinton.
Thomas Jefferson refused the subpoena because the travel wasn't possible.
Richard Nixon was subpoena duces tecum for tapes pertaining to his involvement in Watergate, Nixon refused and challenged it in court which ended in the 8-0 decision US vs Nixon where Nixon was ordered to turn over his tapes. Nixon argued that he had article 2 protections but the courts ruled a President wasn't immune from criminal charges. Nixon resigned before he was subpoena ad testificandum (called to testify).
Bill Clinton was subpoena ad testificandum, but agreed to testify voluntarily which prompted Ken Starr to withdraw his subpoena.
So there isn't any history about what happens if a President just outrights refuses to follow a subpoena ad testificandum but there have been cases where Presidents have been subpoenaed.
In the context, your question provides the sitting President has and can be subpoenaed for something such as a tax return although it isn't a rule for them to do so.
There's no doubt that Donald Trump is a polarizing figure, which could lead to selective prosecution that could be a possible roadblock to subpoenaing tax returns, the prosecution has to show that they have evidence of an underlying crime, or that it will aid them in a criminal investigation, they can't just subpoena them for the purpose of releasing them to the public.