Trump has said he demands loyalty from those in his administration. That is not really true. A person can be loyal, as Tillerson was, and still disagree with actions he thinks is wrong. And Tillerson did disagree with Trump.
He recognized that Iran was abiding by the treaty the two countries had signed, while Trump is looking for any excuse to toss out that treaty. Tillerson was also willing to call Russia to task for its actions (like their poisoning of a man and his daughter in Great Britain), while Trump refuses to acknowledge any wrongdoing by Russia.
The truth is that Trump doesn't want loyalty -- he demands sycophancy. He's not just a narcissist. He takes narcissism to dizzying new heights. He wants his employees to be sycophants -- "yes-men" who will praise whatever he says or does, and praise him again when he says or does the opposite thing a few days (or few hours) later.
Of course, that is why the White House is in chaos. Even the yes-men working there don't know where their boss stands. They would love to defend his ridiculous words and actions, but are afraid that he will change his ming as soon as they do that.
Tillerson was not a great Secretary of State, but that wasn't why he was fired. He was fired because he failed Trump's demand for sycophancy.