The vote is being held to fill a few dozen vacant seats and follows months of political reforms.
Although Suu Kyi would have little say in a parliament still dominated by the military and military-backed ruling party, her entry after decades of struggle against the regime would be hugely symbolic.
The international community is paying close attention to the election as the country continues on its path of reform.
The by-election follows a truce signed with rebels and the release of political prisoners.
Moves towards democratisation may lead to the end of the diplomatic isolation of the last decades and encourage Western powers to ease the economic sanctions that have been placed on the country for years.
Christian Solidarity Worldwide warned last week that human rights abuses were still taking place among ethnic minority communities and Christians.
It has urged the international community to take a cautious approach and lift sanctions gradually.
The warning came after soldiers from the Burmese army reportedly ransacked the Sin Lum Pang Mu Baptist Church in Pang Mu village and disrupted a Christian conference in Sabawngte village.
Benedict Rogers, East Asia Team Leader at Christian Solidarity Worldwide, said that Burma still had a "very long way to go" in its reform process.
He said: "Religious freedom is a fundamental value in any democratic society, and so if the Burmese government is serious about reform, it must protect religious freedom. We urge the international community to monitor the situation closely."