JERUSALEM (AP) _ Israeli soldiers shot and wounded a Palestinian who threw a hand grenade, and prison officials opened fire on protesters who pelted their bus with stones, wounding one woman, the army command said today.

Bethlehem's Mayor Elias Freij told The Associated Press about 300 Palestinian policemen have handed in their resignations, including all 50 officers in Bethlehem.

An army spokesman said the hand grenade did not explode. Soldiers chased the assailant and shot him in the back. He was taken to Hadassah hospital in Jerusalem, but a hospital official said he had no details about the man's condition.

The incident took place Friday evening in the village of Bani Naim, near Hebron in the occupied West Bank. It was the second time Palestinians have thrown hand grenades at Israeli troops since violence broke out three months ago.

Today, prison service employees opened fire on protesters who pelted their bus with stones and broke several windows in the West Bank village of Nur el- Shams, the army spokesman said, speaking on condition of anonymity.

A 50-year-old woman was wounded and taken to a hospital, the spokesman said.

In Bethlehem, soldiers dispersed two demonstrations and arrested several protesters who threw stones and empty bottles at the local police station.

Freij said the resignation of the local police ''would make things very difficult. There was a theft in town about a week ago. Our policemen found the thieves and put them in jail within 12 hours. Who will do this now?''

Freij said all the police in Ramallah and Jericho resigned. The Arab-run Palestine Press Service said 68 policemen in Hebron also quit. The resignations go into effect Monday, Freij said.

On Friday, the army barred Palestinian worshipers from three mosques in Gaza City on the Moslem holy day in an attempt to maintain order. Israeli soldiers wounded nine worshipers after services in the West Bank and Gaza Strip.

Troops razed four Palestinian homes to punish suspects in the attempted lynching of a collaborator and the knifing of a Jewish settler.

Prime Minister Yitzhak Shamir attacked the latest U.S. peace plan and said it was formulated to please the Arabs. The PLO has also rejected the plan because it did not provide for direct Palestinian representation at negotiations.

Troops were posted Friday around Al Amari mosque in Gaza and barbed wire blocked the street to keep out worshipers. Gaza's main mosque draws about 2,500 faithful, and local residents said it was the first time it was closed.

Two other mosques in Gaza City also were shut by the army to prevent demonstrations, which have become routine on Fridays since violence broke out Dec. 8 in the territories seized by Israel in the 1967 Middle East war.

Israeli bullets or beatings have killed 87 Palestinians in anti-occupation rioting.

The army turned journalists away from Nablus and Hebron but the clampdown was not as widespread as the previous Friday, when nearly all the West Bank was closed to coverage.

In Beit Ummar, 10 miles south of Jerusalem, troops shot and wounded eight Palestinians, two of them seriously, hospital officials at Beit Jallah and Mukassed said.

Soldiers fired tear gas canisters to disperse protesters outside the Beit Ummar mosque. Witnesses said at one point troops entered the building to broadcast a message over the loudspeaker ordering protesters to go home.

''The soldiers entered the mosque while we were still kneeling barefoot on our prayer rugs. People started shouting Allahu Akbar,'' or ''God is Great,'' said Youssef Hassan Khalil, 30, who lay on a cot in Mukassed hospital with a wounded right ankle.

In Jabaliya refugee camp near Gaza City, soldiers fired on demonstrators outside a mosque, wounding one Palestinian in the leg, officials at Shifa hospital said.

Israeli troops destroyed one Palestinian home in Hebron and three in Biddya village, 30 miles west of Jerusalem.

The army command said the homes in Biddya belonged to families of three residents who led a lynching attempt March 5 against a man suspected of collaborating with Israel.

Shamir broke his silence two days before flying to Washington and criticized the peace initiative proposed by Secretary of State George P. Shultz.

''The only word I agree with in the U.S. plan is the name Shultz,'' Shamir was quoted as saying in an interview with the daily Haaretz.

The PLO has said that it would not agree to indirect representation through a joint Jordanian-Palestinian delegation as suggested in the Shultz plan.