Last week, the national carrier cancelled flights to three other airports in the northern region.
The escalation in violence in Amhara prompted Prime Minister Abiy Ahmed's government to declare a six-month state of emergency there last week.
"Flights planned for Wednesday, Thursday and Friday to Dessie (Kombolcha), Gondar, Lalibela and Bahir Dar have been cancelled," Ethiopian Airlines said.
The fresh unrest in Africa's second most populous country comes just nine months after the end of a devastating two-year war in the neighboring region of Tigray.
That war also drew in fighters from Amhara.
Tensions have been rising since April, when the federal government announced it was dismantling regional forces across Ethiopia.
That triggered protests by Amhara nationalists who said the move would weaken their region.
The clashes have prompted travel warnings from foreign governments.
Residents of several towns in Amhara told AFP that fighters from the local Fano militia were facing off against federal forces.
Tesfahun, a resident of Bahir Dar, told AFP on Tuesday he had heard artillery fire "non-stop around the airport roads" until midnight on Monday.
"This morning there were gunshots around Lideta 14," a neighborhood near the airport, he said.
"People are indoors. (There is) no movement."
Simachew, a rickshaw driver in Gondar, told AFP that the city had witnessed "heavy fighting this afternoon".
"The outskirts of the city are controlled by Fano and ENDF (Ethiopian National Defense Force) has controlled the higher grounds ... and is firing heavy artillery from there towards the town," he said.
"Fighting is appearing to intensify as more Fano militia are mobilizing from nearby towns."
'We were afraid'
Tourist Inma De Blas said she was part of a group of 19 Spaniards stranded between Bahir Dar and Gondar.
While they were on the road, "somebody told our driver to stop the van," she told AFP by phone.
"Suddenly we heard a lot of shooting, we saw people putting big stones and wood on the road to block it. We were afraid," she said.
When the gunfire eased, they made their way to a small hotel, where they are awaiting further guidance from the Spanish embassy.
"We cannot go to Bahir Dar, we cannot go to Gondar, we cannot go anywhere," their Spanish guide told AFP.
Two residents of Lalibela, a UNESCO World Heritage site famous for its 12th and 13th-century rock-hewn churches, told AFP that the army appeared to be approaching the city, which was taken over by Fano fighters last week.
"The ENDF troops are firing on the town and we are staying inside," said Ayalew.
"Things are getting a little more scary, no one is out," said another resident Aneley, who added that he "didn't see the Fano militiamen in town this afternoon."
The authorities said at the weekend they had arrested people responsible for "the security crisis" in Amhara.
The violence has made humanitarian operations difficult in the region, according to the World Health Organization.
The United States has "expressed concern" about the clashes, while Australia, Britain and Spain have advised their citizens against travelling to Amhara.