Nope. It's a sort of running gag sometimes parroted whenever someone wants to add anti-US or anti-Iraq points to a conversation. Like dozens of things said about Iraq which are either twisted or exaggerated to an entirely different meaning or plain fabricated, this claim has no bearing other than the occasional social media mention or a handful of infamously disingenuous pro-invasion writers in the 1990s-2000s. One claimed Iraq was behind 9/11 among other absurdities, another admitted his book to being a slander work full of stories and rumors with no basis in fact. The well-known lies about being behind AQ and 9/11 and the WMDs including nukes are only the tip of the iceberg.

In 1963, Saddam Hussein was in Egypt. He didn't play a role in the coup. He returned to Iraq after the coup where he was promptly arrested and imprisoned for several years. There is no evidence linking the CIA to the coup. They had an interest in having overthrown Abdul Karim Qassim, but were only planning in the same way they planned and still plan to overthrow most governments in the world. What we know from plenty of declassified documents and plenty of other information over the decades is the CIA didn't play a role in the coup.

However, by this point the Communist-leaning Abdul Karim Qassim had alienated every cross-section of Iraqi society. When the Communists were his last supporters, he had them largely killed too. Due to how underdeveloped Iraq was and the general lack of global interest, we know little of the extent of the Stalinist style repression coupled with a very weak government, but needless to say, Qassim was ruling on borrowed time, and his engagement in a massive civil war against militants, backed by Iran, the US, and Israel, killing tens of thousands of Iraqis on both sides and scorching the Kurdish countryside didn't help his consolidation of power.

The origin of this "CIA coup" claim comes from King Hussein of Jordan. When he was accused of being an American agent and puppet, his response was to accuse America of being behind the coup in Iraq that happened shortly before. He was also spiteful of Iraq because Abdul Karim Qassim's 1958 coup killed the royal family, who was King Hussein's blood relatives. A benign and worthless statement was said, and unwittingly a myth would be born decades later. It wasn't until after the Gulf War and leading up to and after the 2003 Iraq invasion where this and many other false or twisted claims became regurgitated. It was a ~20 year era where any war hawk could say anything against Iraq and unfortunately it became "fact" despite little to no evidence.