Saul was rejected as King over Israel because he disobeyed God's command to utterly destroy the

 1samuel 15:1-33

  Because you have rejected the word of the LORD, He also has rejected you from being king: In his empty religious practice, rebellion, and stubbornness against God, Saul rejected God’s word. So, God rightly rejected him as king over Israel. Saul was rejected as king not specifically because he offered sacrifices, but because he disobeyed a direct command that God had given him through the prophet Samuel. Samuel had told Saul, “Go down ahead of me to Gilgal. 

I will surely come down to you to sacrifice burnt offerings and fellowship offerings, but you must wait seven days until I come to you and tell you what you are to do.”  But Saul, worried that his whole army would desert him, offered the sacrifices himself, just before Samuel arrived.

“You have done a foolish thing,” Samuel told him. “You have not kept the command the Lord your God gave you; if you had, he would have established your kingdom over Israel for all time. But now your kingdom will not endure; the Lord has sought out a man after his own heart and appointed him ruler of his people, because you have not kept the Lord’s command.”  In other words, the penalty for this outright disobedience to a direct command from God was that Saul would not be the founder of a royal dynasty; while he would remain king, his descendants would not rule after him.

Secondarily, however, this disobedience did lead Saul to usurp a privilege of the priesthood. By offering these sacrifices, Saul was imitating the Canaanite priest-king model instead of respecting the separation between the kingship and the priesthood that was established in the law of Moses.

Saul subsequently disobeyed another direct command from God when he was told, again through the prophet Samuel, to completely destroy the Amalekites.*  Saul instead kept their king, Agag, alive as a trophy of war, and his soldiers kept the best of the cattle to “sacrifice to the Lord”—as part of a grand feast that they would enjoy themselves.  Samuel asked Saul once again, “Why did you not obey the Lord?”  The penalty for outright disobedience this time was that Saul would not even remain king himself for his natural lifetime; he would die early and be succeeded by “one of his neighbors”—not one of his own descendants.