President Trump had no more duty to “obey” an executive order issued by Barack Obama than he was obligated to wear Obama’s underwear that was left behind in the White House.
President Trump ignored prior executive orders whenever it suited him. He went through much of his tenure in office, for example, without nominating someone to fill a vacancy in the Associate AG post — which was never established by Congress, and was instead established through an internal DOJ order in the 1970s.
A President is part of the Executive Branch, and is therefore subject to orders governing it.
Q: How do you end or overturn an Executive Order?
A: The President who issued an Executive Order can revoke it. Likewise, an incumbent President has the power to revoke an Executive Order issued by a predecessor.