Henry Nowak, 18, died after being stabbed by Vickrum Digwa in the English city of Southampton in December.
Digwa, who is Sikh, falsely claimed to police he was the victim of a racist assault by Nowak, who was white.
When police officers arrived, they initially treated the wounded man as a suspect before noticing his injury and trying to resuscitate him.
Digwa, 23, was convicted of murder for stabbing Nowak with a 20-centimetre Sikh dagger and sentenced on Monday to life in prison with a minimum 21-year term.
The case has been seized on by anti-immigration activists and politicians, despite the fact that both Nowak and his killer were British.
On Tuesday, police in Southampton were pelted with chairs, cans, rocks and flares after a demonstration over Nowak's death attended by far-right figures and others.
Vance said in a post on X on Friday that there should be "righteous anger" in response to the murder, which he blamed in part on "the mass invasion of migrants, many of whom despise the West and the people who love it".
In a statement issued in response to Vance's comments, Starmer's office criticised people "trying to interfere in our democracy and seeking to stir up division on our streets".
"The Nowak family are grieving after Henry's horrific murder. They have said they do not want his death to be used to create further division, hatred or tension. We should be respecting their wishes," Downing Street said in a statement.
"Our politics should bring people together even in the most terrible of circumstances. That is who we are as a country."
Politicians including Nigel Farage, leader of the hard-right party Reform UK, have claimed that the police response is evidence of "two-tier" policing, with a bias against white people in the British justice system.
The US State Department echoed the "two-tier" policing claim in a post on X on Thursday, expressing condolences to Nowak's family and saying that "Ideological conditioning and two-tiered policing are glaring symptoms of civilisational decline".
The British government rejected the "two-tier" allegation, which is not backed by statistical evidence.
The Independent Office for Police Conduct, which investigates allegations of police wrongdoing, is probing the actions of police officers responding to Nowak's stabbing.
The victim's father, Mark Nowak, has said the case was not about racism or religion, and that he wanted his son's death to lead to safer streets and not to be used to create "further division, hatred or tension".
