The Supreme Court yesterday rejected the appeal of former Madeira School headmistress Jean Harris, convicted murderer of Scarsdale Diet author Dr. Herman Tarnower.

The action, made without comment or dissent, ends the legal aspects of the celebrated case three years and 11 days after it began with the shooting of Tarnower, a cardiologist, at his Purchase, N.Y., home.

Only some extraordinary new information or legal theory now could keep Harris, 59, from continuing to serve her sentence of 15 years to life in prison.

She has been jailed at Bedford Hills Correctional Institute in New York since her conviction Feb. 24, 1981.

Harris argued during her trial and after it that she intended to commit suicide that day and that Tarnower was killed while trying to take the gun from her.

Prosecutors said the shooting was her jealous reaction to Tarnower's attempt to break off their 14-year relationship.

In her appeal to the Supreme Court Harris contended that her trial was unfair on two grounds. She said the jury should not have been allowed to hear damaging testimony from a police officer about a telephone conversation he overheard between Harris and her lawyer on the day of the shooting.

The officer testified that he heard her say: "Oh, my God, I think I've killed Hy."

She also cited what the appeal called massive prejudicial publicity resulting from the judge's failure to close pretrial hearings in her case to the media and the public.

Harris' appeal of New York state court rulings against her was filed five days after the official Supreme Court deadline because of what her lawyer, Herald Price Fahringer, called "heavy professional commitments."

Prosecutors asked the justices to reject the appeal because it was late. The justices did not explain their action yesterday.

In other actions yesterday:

* The court said it would consider an important death penalty question: whether state appellate courts must review each death sentence to see if it is out of proportion to punishment given others convicted of similar crimes.

The case, Pulley vs. Harris, stems from a double murder and kidnaping in connection with a 1978 bank robbery in San Diego.

Robert Alton Harris was convicted of shooting two teen-age boys whose car he had stolen as a getaway vehicle in the robbery.

The conviction and sentence were upheld in the state courts.

The 9th Circuit U.S. Court of Appeals, however, ruled that state courts must conduct a "proportionality review" before approving the death penalty.

Although the Supreme Court has treated such reviews as useful, it has never required them explicitly and has approved death penalty statutes that did not require them.

* The court declined involvement in a controversy about a California "report-a-crime" law requiring psychotherapists and others to report alleged child abuse, even when it means breaching patient confidentiality.

The case, Forbes vs. Newman, began when a therapist told authorities about an incident he had learned from a man during a private session.

The man was acquitted of child abuse after the episode but not before temporarily losing custody of his children.

He had asked the justices to require that patients be warned that anything they say may be used against them under such laws.

* The court agreed to review 1977 amendments to the Social Security Act that treat men and women differently in pension eligibility.

The case, Heckler vs. Mathews, grew out of Congress' attempt to protect some government workers from a 1977 Supreme Court ruling declaring such gender-based distinctions unconstitutional.