In Birchfield v. North Dakota, the U.S. Supreme Court held that laws that make refusing a DUI blood test a crime are unconstitutional. (579 U.S. ___ (2016).) The Court also held that an officer can't require a driver to submit to a blood test without first getting a warrant.
The Birch decision made clear, however, that officers conducting legitimate DUI arrests don't need warrants to demand that drivers complete breath tests. And it clarified that laws can criminalize breath-test refusal without running afoul of the Fourth Amendment.
For much more on the decision, including its effect on civil (or "administrative") penalties for refusing a blood test and the potential that the law differs somewhat from state to state, see DUI Testing: Breath, Blood, and Warrants.