The death of Lebanese President-elect Bashir Gemayel sent tremors through official Washington last night and touched off speculation about the potential adverse effects on President Reagan's hopes of restoring peace in Lebanon and then moving to resolve the Arab-Israeli conflict.

The administration's concern about the impact on its top-priority Middle East policy goals was underscored by the silence with which U.S. officials initially reacted to reports of Gemayel's death. For several hours last night, the White House and the State Department turned aside all questions by saying they had no confirmation and thus were unable to comment.

Late last night the White House issued a statement condemning the "cowardly assassination" and promising "full support" to Lebanon "in this hour of need."

"The tragedy will be all the greater if men of goodwill in Lebanon and in countries friendly to Lebanon permit disorder to continue in this war-torn country," the president said.

Secretary of State George P. Shultz was conferring urgently with his senior Mideast advisers to assess the damage and study possible fallback positions to maintain momentum in U.S. efforts to defuse the Lebanese crisis. Although there were no immediate signs of what the United States might do specifically, the problems posed by Gemayel's murder were obvious.

Although the 34-year-old Christian leader was hated by leftist Moslem groups in the faction-ridden country, he had been maneuvering since his election as president 23 days ago to achieve a rapprochement with the Moslems and restore the delicate political balance that had become engulfed in civil war over the last decade.

Only last Friday, Shultz, in testimony before the Senate Foreign Relations Committee, said Gemayel was "a strong and effective individual and he has a very good chance of becoming a strong and effective leader. . . . " As some administration sources candidly conceded last night, there is no one else in sight who seems capable of filling that role effectively.

Of most urgent concern, U.S. officials noted, was whether there would be new tensions between Lebanon's Christian and Moslem communities that would derail attempts to restore the power and authority of the central government and the companion effort to bring about withdrawal of all foreign forces -- Israeli, Syrian and Palestine Liberation Organization -- from the country.

Although responsibility for the bombing was unknown, there were fears here that Gemayel's followers in the Christian Phalange Party would blame the Moslems and seek revenge in a new outbreak of the civil war that has left Lebanon shattered and with large parts of its territory under armed Israeli or Syrian occupation.

Even if a new bloodbath is averted, U.S. officials are worried that renewed animosities between Christians and Moslems might cause a breakdown in U.S. efforts to mediate between the two sides. Gemayel's death came as U.S. special ambassador Morris Draper had returned to the Middle East to seek ways of strengthening the central government and of inducing the Israelis and the Syrians to withdraw before their increased clashes of recent days escalate into full-scale warfare.

As various officials noted last night, Israeli Prime Minister Menachem Begin's government, which had counted Gemayel as an ally, now seems likely to conclude that the Lebanese situation still poses acute dangers to Israel's security and requires a continued massive Israeli military presence.

Similarly, the Israelis could renew their demands that Lebanon sign a peace treaty as a condition of Israel's withdrawal. The Israelis had appeared to be backing away from that demand under prodding from Shultz, who recognized that it was making more difficult Gemayel's efforts to conciliate the Moslems; but the new uncertainties could force Jerusalem back into an uncompromising stance.

It also was apparent that the assassination could have ripple effects extending beyond Lebanon and seriously affecting the prospects for progress on Reagan's Mideast peace initiative calling for a freeze on Israeli settlements in the West Bank and Gaza Strip and eventual self-rule for these territories "in association with Jordan."

Much of the impetus for the initiative stems from the U.S. view that with the expulsion of the PLO from Beirut and progress toward stability in Lebanon, Israel would no longer need to fear for the safety of its northern borders and might be more willing to negotiate the surrender of the territories it has controlled since 1967.

Now, several sources noted, there is certain to be a setback to efforts aimed at getting Israel to turn its attention away from Lebanon and focus on other questions. At the same time, the sources added, the Gemayel killing is a reminder that strong hatreds still exist in the region and that these passions might be seized upon by radical Arab forces to warn their more moderate brethren among the Palestinians and in other countries that they risk dying if they deal with Israel.