Three days after his nomination to the Fairfax County School Board, William W. Delaney turned down the Mount Vernon District seat, saying the demands on his time would prevent him from meeting his family obligations.

Mount Vernon Supervisor T. Farrell Egge, who announced the nomination last Thursday, yesterday appointed Olivia Michener, 43, to fill the vacant seat on the School Board. Michener was sworn in at noon at the school administration headquarters in Fairfax City.

In a letter to Egge, Delaney, 47, an economic consultant, said that "the same parental instincts that led me to seek the position at this critical time for our schools have, upon further consideration, persuaded me that the likely impacts on my family would be unacceptable."

Egge, a Republican whose election gave the Board of Supervisors its first GOP majority in more than a century, made the appointment to fill the seat vacated by Gerald A. Fill. Fill, who was appointed by former Supervisor Sandra Duckworth, stepped down after Egge's election.

Michener was a member of the citizen task force that studied declining enrollment in the county and produced the guidelines for determining what conditions should be present before a school is targeted for closing. She takes over the job at a time when the School Board must decide whether to close any of the three high schools in the eastern part of her district, where enrollments have declined steadily. The schools are Groveton High, Mount Vernon High and Fort Hunt High.

Superintendent William J. Burkholder has recommended closing Groveton as a high school and turning it into an intermediate school. A task force of citizens had previously recommended closing nearby Fort Hunt and making it an intermediate school. In each case, the recommendations prompted strong opposition from the communities around the schools.

Michener said yesterday she would try to solve the problem of declining enrollments in a way that preserved all three high schools.

Michener earned a bachelor's degree in education from Goucher College in Towson, Md. She moved to Fairfax County in June 1973. Her husband, Frank, is a psychiatrist specializing in children and adolescents. She has three children.