How can US Supreme Court decisions be overturned?

No single entity - not the President, Senate, House of Representatives, state Governors, nor anyone else - has the power to overturn a US Supreme Court ruling. Supreme Court decisions cannot be nullified by other parts of government. However, if the Supreme Court strikes down a federal law, Congress can always modify the law until it is such that the Supreme Court does not consider it to violate the Constitution, then pass it again.

Supreme Court decisions can only be overturned in two ways:

Legitimate Methods
  1. The US Supreme Court reverses a decision on an earlier case by making a contradictory decision on a current case.
  2. Congress and the States can overturn a decision by amending the Constitution.

Illegitimate Methods (Passive Resistance)
  1. Sometimes the Executive Branch obstructs or fails to enforce a decision.
  2. Sometimes Congress rewrites legislation to bring it into compliance with constitutional guidelines.
  3. Sometimes Congress strips the Supreme Court of its appellate jurisdiction over certain types of cases to deprive them of the ability to overturn a law or policy.
  4. Sometimes states pass laws that clearly violate Supreme Court decisions, forcing someone with standing to challenge the new law's constitutionality. Meanwhile, the law can be enforced even if violates established civil rights. State legislatures do this with the hope of overturning, or slipping around, precedents set by earlier Courts.

Contributor's Example
Don burns the US Flag on the steps of a state capitol. The state arrests him because the state amended its own constitution to make this illegal. However, the US Supreme Court ruled that flag burning is protected speech, thereby making flag burning lawful under US law.

If the case is appealed to the US Supreme Court, it would rule that the state constitutional amendment violates the US Constitution (which is superior) and strike down the state amendment.

The state then writes a law that Don has to buy a permit to burn the flag in any form of protest. The US Supreme Court might uphold that law (if challenged) as long as it fulfills a legitimate government purpose, but does not: 1) discriminate in who the permits are issued to, or 2) make the cost or time involved in issuance of the permit unbearable. Sometimes such laws are deemed unconstitutional, and sometimes not.