The army is aiming to end the months old siege but lost its 100th soldier while doing so and another two were killed overnight, security sources said.
At least 81 Fatah al-Islam fighters and 40 civilians have also been killed in the worst internal violence since Lebanon's 1975-1990 civil war.
The advance marked a major step for the army in the battle to crush fatah al-Islam and a rare incursion by troops into a Palestinian refugee camp in Lebanon.
A 1969 Arab agreement banned Lebanese security forces from entering Palestinian camps.
The agreement was annulled by the Lebanese parliament in the mid-1980s but the accord effectively stayed in place.
Commandos exchanged automatic rifle fire and grenades with fighters at buildings and alleyways leading to the centre of Nahr al-Bared while army artillery and tanks pounded other areas.
The vast majority of Nahr al-Bared's 40,000 refugees had fled in the early days of the fighting which has left the camp in ruins.
The Unifil vehicle damaged near Tyre was hit by an explosion near the Qasimiya bridge over the Litani river.
"We can confirm an explosion in the area of Qasmiyeh bridge which slightly damaged a Unifil vehicle of the Tanzanian battalion," spokeswoman Yasmeena Bouziane said.
The blast came less than a month after a car bombing cost the lives of three Spanish and three Colombian members of Unifil on June 24.