Israeli troops have pushed deeper into southern Gaza, a day after the United Nations condemned Israel for killing at least eight Palestinian protesters.
Many Rafah families are in mourning
Helicopter gunships fired missiles as troops moved into two areas on the south side of the town of Rafah.
Eight Palestinians were reported killed in Rafah refugee camp, as the victims of Wednesday's violence were buried.
Aid officials warn of a crisis in Tel Sultan - the Rafah district that has born the brunt of the incursion.
More than 40 Palestinians have died since Tuesday when Israel launched one of its biggest operations in Gaza in decades.
Last week,13 Israeli soldiers were killed by Palestinian militants in the Rafah area.
On Thursday, Israeli forces moved into two southern districts of Rafah, trading fire with gunmen, in the search for tunnels used to smuggle weapons.
The local leader of the radical Hamas movement died in an Israeli missile strike, Palestinian medical sources said, taking the day's Palestinian death toll to eight.
Mourners surged through the streets of Rafah in a chaotic procession to the cemetery to attend the funerals of those killed a day earlier.
Palestinians said at least eight Palestinians died on Wednesday when an Israeli tank opened fire on demonstrators protesting against the demolition of homes in Rafah. Israel put the death toll at seven.
In Rafah on Thursday, people also carried milk, water and food that they were hoping to get to the people of Tel Sultan, which has been occupied by Israeli troops for the past three days.
DAYS OF VIOLENCE
Tues: Israeli raid begins
Wed: Protest march - eight Palestinians die
Thurs: Eight Palestinians killed
Total Palestinian deaths: 41
Relief groups say supplies are running dangerously low and warn of a humanitarian crisis unless water and electricity are restored to the district's 25,000 inhabitants.
The BBC's Alan Johnston says 22 Palestinians have died in Tel Sultan, but relatives cannot bury them until the curfew is lifted.
Rafah's mortuary, he adds, has overflowed and many of the dead have had to be stored in freezers in different parts of town.
Israel has vowed to continue its operation to find wanted militants and demolish houses concealing tunnels used to smuggle weapons from Egypt.
Following Wednesday's violence, the UN Security Council adopted a resolution condemning the deaths and demolitions.
The resolution, which was adopted by 14 votes to 0, also urged Israel not to demolish homes in violation of international law.
The US, which usually vetoes anti-Israeli resolutions, abstained from the vote.
The White House issued a statement, saying it did not believe Israel's actions in Gaza served "the purposes of peace and security".
The UN says recent house demolitions in Rafah by the Israeli army have left about 1,600 residents homeless.
The Israeli army apologised for the Palestinian deaths but said troops did not deliberately fire on marchers.