The security forces were hunting for those thought to be behind the violent riots that erupted two days ago.
Residents were angered that the budget, presented in parliament on Wednesday, confirmed the construction of a gas pipeline from Mtwara to Dar es Salaam.
They want a gas processing plant to be built in Mtwara instead.
The 500km (310-mile) pipeline is part of a $1.22bn (£807m) project which will also see the construction of a gas processing plant in Dar es Salaam, the main city of the East African nation.
Fleeing
Journalist Abdallah Bakari, who works for Tanzania’s privately owned Mwananchi newspaper, told the BBC Swahili service that the woman was shot in her home on Thursday.
The security forces were going house-to-house to search for people they thought had been involved in the clashes with police.
Dr Mohamed Kodi at Mtwara’s Ligula Referral Hospital told Mwananchi that the woman who died was shot in the stomach. She was seven months pregnant, he said.
Her shooting was also reported in the Kiswahili-language paper Nipache.
The local police commander has refused to comment on the incident.
Correspondents say many men have fled the town because of the crackdown and some women and children have sought shelter in the grounds of Ligula Hospital for safety.
When the riots broke out on Wednesday, some buildings and vehicles were set alight, cars were stoned and a bridge blocked to stop traffic.
Riot police were deployed and soldiers can now be seen patrolling the streets.
Police have been escorting buses travelling into Mtwara to stop them being attacked.
Shops, schools and other businesses in the town remain closed because of the tensions.
There were similar protests in January about the proposed pipeline when the houses of several governing party politicians were burnt down.
Parliament has been temporarily suspended because of the latest trouble.
But President Jakaya Kikwete has said the government will continue with the project despite the protests.
Their demands are not valid because resources belong to the whole country, not one geographical area, he said.