The Palestinians have been holding daily demonstrations at the border after Israel closed the Erez crossing from Gaza.
It is the only gateway for pedestrians entering Israel.
The closure has prevented thousands of workers from the impoverished enclave from entering Israel.
During clashes on Monday, two protesters were wounded by "Israeli bullets", the Hamas-controlled Health Ministry in Gaza said in a statement.
The army, meanwhile, said a drone "struck a military post belonging to the Hamas terrorist organisation".
The post was near an area from which assailants shot at soldiers "during a violent protest", an army statement said.
The "soldiers responded with live fire toward an assailant … and hit him".
On Monday, protesters burned tyres and threw rocks at soldiers who responded with tear gas, AFP reported.
Hamas puts weapons and ammunition on display in Gaza - in pictures
Since September 13, seven Palestinians have been killed and dozens wounded in violence at the border, according to figures from the Health Ministry in Gaza.
Israel has imposed an air, land and sea blockade on the Palestinian enclave since the Islamist group Hamas seized control in 2007.
Armed conflict sporadically erupts between Israel and militants in the Gaza Strip.
In May, an exchange of Israeli air strikes and Gaza rocket fire resulted in the deaths of 34 Palestinians and one Israeli.