Some of the opposition National League for Democracy party members had been attending a ceremony at party headquarters to mark General Aung San's death 62 years ago. Others had been at the official commemoration ceremony at Martyr's Mausoleum.
At least 50 NLD members were walking in small groups Sunday when they were arrested, witnesses said on condition of anonymity for fear of official reprisal.
It was not immediately clear why police detained them.
"Some members were roughly taken into trucks, and those who ran away were chased," a witness said.
Some who ran onto public buses were dragged out and taken away.
At least three journalists and cameramen who had been filming NLD members walking to the mausoleum were detained.
They were released minutes later after police told them not to use video footage that showed heavy security.
General Aung San and other government leaders were assassinated by gunmen during a Cabinet meeting on 19 July 1947, shortly after Britain granted independence to the Southeast Asian colony.
Nobel Peace Prize laureate Aung San Suu Kyi marked the anniversary of her father's death inside Yangon's Insein prison.
She is on trial on charges of violating the terms of her house arrest by giving shelter to an uninvited American man who swam to her lakeside home in May. If convicted, she faces up to five years in prison. Her trial is to resume Friday.
Earlier on Sunday, hundreds of riot police erected barricades secured with barbed wire and blocked streets leading to the Martyr's Mausoleum.
More than two dozen trucks carrying riot police and four prison vans were parked near the monument, located near the famed Shwedagon pagoda.
Flags were flown at half-staff at the mausoleum as officials placed flowers at the tomb, and families of the slain leaders joined the tightly guarded wreath-laying ceremony.
Authorities allowed about 30 NLD members to pay tribute at the mausoleum but turned away others who wore T-shirts emblazoned with images of Aung San.
Suu Kyi, 64, who used to attend the official ceremony, was absent for a sixth consecutive year and instead marked the day by donating food to patients at the hospital inside the prison, said Nyan Win, a spokesman for her party.
Martyr's Day was an important event on Myanmar's calendar for years, but has been gradually downgraded as Suu Kyi has become more popular, particularly since a 1988 pro-democracy uprising that was crushed by the junta.
Myanmar, also known as Burma, has been under military rule since 1962.
Suu Kyi has been under detention for 14 of the past 20 years. Her opposition party won national elections in 1990, but Myanmar's generals refused to relinquish power.
Her trial has drawn condemnation from the international community and her supporters within Myanmar, who worry that the ruling junta has found an excuse to keep her detained through elections planned for next year.