Retired Adm. Elmo R. Zumwalt's appointment to an arms control advisory board appears certain to die with the lame-duck session of Congress now that Virginia's senators have insisted that he be quizzed whether the job would place him in a conflict of interest because he writes a newspaper column.

Sen. John W. Warner (R) warned the Senate Wednesday that the Zumwalt nomination constituted "a very important precedent" and asked that the former chief of naval operations submit to questions from the Foreign Relations Committee.

Sen. Charles H. Percy (R-Ill.), the committee chairman, has not said whether he will hold hearings on Zumwalt's appointment to the General Advisory Committee of the Arms Control and Disarmament Agency but, given the short time left in the lame-duck session, most observers said the nomination is all but dead.

An arms control agency spokesman said yesterday the Reagan administration has no plans to back off the nomination. Zumwalt, a Democrat, ran against Sen. Harry F. Byrd Jr. (I-Va.) in 1976 and supported Reagan in 1980.

To keep the nomination alive, the White House will have to submit it anew when the new Congress convenes in January. "We are still hopeful we can get a confirmation," said Joseph Lehman, the arms spokesman.

Zumwalt's nomination cleared the Senate last fall before either Warner or Byrd discovered it and urged the White House to return it to the Senate. In an unusual move, the White House agreed. At the time, Warner said that his objections to Zumwalt were not personal. The two served together when Warner was secretary of the Navy and in his memoirs, Zumwalt criticized Warner as an administrator "who bent with every political breeze that blew."

On the Senate floor Wednesday, Warner -- joined by Byrd -- restated his objections to the nomination. He said his concern stemmed from a fear of giving a journalist access to sensitive national security documents. Zumwalt had said previously he would abide by "the same rigid standards I applied while serving on active duty."

In defense of the nomination, Eugene Rostow, arms control director, noted that other journalists have served on the 14-member advisory board. He said it was "unlikely" that Zumwalt's access to information would increase the marketability of the syndicated column that Zumwalt writes with Worth Bagley, another retired admiral.