* Protesters beat up police officer they say shot at them
* Demonstrations now in third week
* Constitutional court says president has right to run
* European donors suspend some aid to Burundi (Adds details from Reuters witness, reports of three deaths)
By Goran Tomasevic and Patrick Nduwimana
BUJUMBURA, May 12 (Reuters) - Police fired guns and teargas towards protesters throwing stones in a suburb of Burundi's capital on Tuesday during a demonstration against the president's bid for a third term, Reuters witnesses said.
At least two police officers fired guns towards lines of demonstrators during clashes in Butarere district, one Reuters witness said. Another saw protesters grab a policewoman and beat her up, saying she fired at them, before letting her go.
A protester told Reuters a woman was shot dead in Butarere, where there was heavy gunfire. Local media, citing police, said a grenade thrown in another district killed two people.
A police spokesman had no immediate comment, but police officers have regularly denied shooting at protesters.
If confirmed, the deaths would take the number killed since demonstrations erupted on April 26 to at least 22, based on an unofficial tally by activists.
Opponents say President Pierre Nkurunziza's bid for another five years in office in a June 26 election violates the constitution and a peace deal that ended civil war in 2005. Both documents set a two-term limit but a court ruled he could run.
"We want good leadership. We are suffering every day," one protester told Reuters Television, without giving his name. "We don't want his third term. The police men are killing people."
The unrest has plunged the poor African nation into its worst crisis since the end of a conflict a decade ago that pitted rebel groups of the majority ethnic Hutus, including one led by Nkurunziza, against minority Tutsis, who led the army.
East African leaders, including Nkurunziza, will meet in Tanzania on Wednesday to discuss the crisis and rising tensions in a region with a history of ethnic conflict.
More than 50,000 people have fled Burundi to neighbouring states including Rwanda, where a genocide killed 800,000 people in 1994.
The violence has drawn strong rebukes from Western donors, on whom Burundi depends to finance much of its budget. The United States, which provides support to the army, has demanded police stop using "violent force" against protesters.
Protesters in Butarere, a suburb where there have been frequent rallies, built barricades in the roads, using stones and burning tyres. Local media said they had tried to block the airport road but police stopped them.
Till now, protests have shown no signs of spreading significantly to other cities or rural regions, although local media have reported sporadic demonstrations in a few locations.
Nkurunziza has said he would press on with his election bid, although the United States, other Western nations and several African countries have urged him not to run.
The European Union and others have suspended or are reviewing some aid, saying violence must stop.
Nkurunziza has pointed to a constitutional court ruling that said his first term did not count because he was picked at the time by lawmakers not chosen in a popular vote.
Opponents say the court is biased and Britain, also a donor, has questioned the court's neutrality.
(Additional reporting by Njuwa Maina and Reuters Television; Writing by Edmund Blair; Editing by Andrew Heavens)