There is a longstanding tension between the State Department and the CIA, which frequently plays out within individual embassies between ambassadors/deputy chiefs of mission (who usually runs day-to-day embassy operations) and the CIA station chiefs who run intelligence operations in-country.

The fact that intelligence officers under 'legal' cover in foreign embassies are usually accredited as diplomats, with some ostensibly legitimate and suitably innocuous job title (think 'deputy agricultural attaché'), can breed resentment between them and the 'real' diplomats. The perennial fear of the foreign service professional is that your resident spook gets caught in an embarrassing or illegal situation — which can have a long-term impact on the embassy's diplomatic work.

/u/extremelyinsightful is right that the standard operating procedure (for most intelligence agencies around the world) dictates that the ambassador/diplomatic staff be briefed and sign off on intelligence operations — but I'd take that with a big pinch of salt. It's a complex relationship, because the station chief is the servant of two masters: on the one hand, they (nominally) work for the ambassador within the embassy; in the larger context, however, they work for Langley.

In any case, there have been plenty of times (especially in situations where the relationship between embassy personnel and the CIA station has deteriorated) where State Department personnel being left out of the loop. Tim Weiner's scathing history of the CIA, Legacy of Ashes, offers several examples of these kinds of protocol breaches. Here's an example from Egypt in the 1960s:

"Once again CIA operations have created a major problem," the State Department's Egypt desk informed Luke Battle, the new assistant secretary of state for the Near East. Egypt's ruler, Gamal Abdel Nasser was complaining — not for the first time, and not without cause — that the agency was trying to overthrow his government. "CIA appears to hope that these incidents can be swept under the rug," said the message to Battle. "This should not be allowed to happen."

Battle knew what the CIA's work in Egypt entailed. He had been the American ambassador when a happy-go-lucky case officer carelessly exposed the agency's relationship with a prominent Cairo newspaper editor named Mustafa Amin. Amin had been close to Nasser; the CIA paid him for information and for publishing pro-American news reports. The Cairo station chief had lied to the ambassador about the agency's relationship with Amin. "He had been on the U.S. payroll," Battle said. "Bruce Odell [the CIA case officer] had been meeting regularly with Mustapha Amin. I had been assured that no funds had been exchanged in Egypt, but a photograph of such a transaction was made when Mustapha Amin was arrested."

Some general examples from South America in the 1970s:

In Mexico, the president dealt exclusively with the station chief, not the ambassador, and he received a personal New Year's day briefing at his home from the director of central intelligence. In Honduras, two successive station chiefs had privately pledged the support of the United States to the military junta, in defiance of the ambassadors they served.

And a specific example from Chile, and the CIA's notorious efforts to keep Salvador Allende from assuming the presidency in 1970:

The CIA divided the Allende operation into Track One and Track Two. Track One was political warfare, economic pressure, propaganda, and diplomatic hardball. It aimed to buy enough votes in the Chilean Senate to block Allende's confirmation. If that failed, Ambassador Korry planned to persuade President Frei to create a constitutional coup. [...]

Track Two was a military coup. Korry knew nothing about it.

[...]

On October 6, one of the false-flaggers had a long talk with [Chilean general Roberto] Viaux. Within hours, Ambassador Korry learned for the very first time that the CIA was plotting a coup behind his back. He had a screaming confrontation with [Santiago station chief] Henry Huckster. "You have twenty-four hours to either understand that I run you or you leave the country," the ambassador said.

And one final example, from 1970s Greece:

In the spring of 1974, General Demetrios Ioannidis took over as leader of the junta. He had been working with the CIA for twenty-two years. The agency was Ioannidis' sole contact with the government of the United States; the ambassador and the American diplomatic establishment were out of the loop. Jim Potts, the CIA station chief, was the American government, insofar as the junta was concerned.

(It's outside the 20-year rule for this sub, but if you do some googling around the post-9/11 relationship (particularly around the CIA's paramilitary expansion and drone war, and the Benghazi embassy attack), you'll find plenty of contemporary discussion of the tortured relationship between Langley and the State Department.)

Edit: typos.

All covert action within a country requires approval of the local Ambassador, full stop. The only question is really how much oversight the Ambassador chooses to place on CIA operations. So yes, the local ambassdor ALWAYS is briefed and must give approval.

Also, all covert actions are approved by the U.S. President at National Security Council (NSC) meetings. The Secretary of State is a key permanent member of the NSC, and thus would also be aware of all approved covert actions.

Source: Cryptome hosts one of the now unclassified coordination SOP from 1967 - http://cryptome.org/2012/06/cia-covert-approval.pdf

In the case of Chile, given the build up to then pending lawsuit scheduled for 9/11/2001, the CIA published a full internal investigation. The US Ambassador (and the Department of State in general) was very involved in covert actions for over a decade leading up to the events of 1973.

https://www.cia.gov/library/reports/general-reports-1/chile/

During the Eisenhower years, given that the head of the CIA and the Secretary of State were brothers (Allen and John Foster Dulles), the two agencies worked together to a surprising degree. While I wouldn't go as far as Kinzer did recently in his book "The Brothers" (Kinzer argued that Allen waited for Foster to pick the targets and then went to work), the Dulles brothers did work together, and Foster would remove or relocate any State officials who disagreed with the necessity of a coup. Foster is known to have done this at least twice, in Guatemala and Iran, when local State officials argued no coup was necessary, or that a coup would cause more damage than it would fix. These officials were quietly relocated and replaced with ones more amenable to regime change experiments.

This isn't to suggest the two, or their agencies worked in harmony. CIA officials often disagreed with State positions and as time went on saw more nuance and subtly in the communist world. This was, to a greater or lesser extent reflected by their bosses.

While neither brother ever adopted a progressive view of communism, Allen came to recognize that anti-American actions were not necessarily communist acts. Allen Dulles eventually realised and accepted that some peoples and nations had actually suffered due to the U.S.'s rise to dominance, whereas Foster was almost incapable of seeing anything anti-American as anything other than communist.