The shadow attorney general apologised for the message, which showed a terraced house with three England flags, and a white van parked outside.
UKIP said she had "sneered, and looked down her nose at a white van in Strood with the cross of St George on it".
Labour leader Ed Miliband was "angry" at her, a senior figure told the BBC.
The resident of the house, Dan Ware, said Ms Thornberry - the MP for Islington South and Finsbury - was a "snob".
"I've not got a clue who she is - but she's a snob," he told the Sun. "We put the flags up for the World Cup (in 2014) and will continue to fly them."
Mr Ware, a car dealer, said he would never vote for Labour in the future, adding that it did not "matter" who was in government.
"I think they (Labour) need to get out of their mansions and visit the working class. Her and Ed (Miliband) should come and say sorry to me."
Ms Thornberry posted the image on Thursday, while voting was taking place in the by-election in Kent. Alongside the picture, she wrote: "Image from Rochester."
Labour came third in the high-profile poll behind UKIP, which won the seat and saw its second MP elected to Westminster.
Speaking outside her London home, Ms Thornberry said she had "made a mistake" and apologised "if she had upset or insulted anybody".
Shadow Foreign Secretary Douglas Alexander said Labour leader Ed Miliband had "not held back" in expressing his dismay with the MP's actions.
"Anyone who wants to stand for election and be successful next May has to start with a fundamental and deep respect for voters," he told BBC Radio 4's Today.
"The anger Ed (Miliband) felt when he saw that tweet reflected his understanding that we need to earn the support of people around the country."
BBC political editor Nick Robinson said the tweet had "given the Tory press an alternative narrative" to the party's defeat in Rochester.
He said: "It is the most extraordinary self-inflicted wound I have seen an opposition party inflict on themselves in many, many years."
Prime Minister David Cameron said the Labour MP's actions were "completely appalling", suggesting that she was "sneering at people who work hard, are patriotic and love their country".
The 54-year-old entered Parliament as MP for Islington South and Finsbury in 2005 and served as shadow energy and health spokeswoman before taking the role of shadow attorney general in 2011.
The daughter of a former assistant secretary general of the United Nations, she was born in Surrey and was called to the bar in 1983, specialising in criminal law.
She had a majority of 3,569 over the Liberal Democrat candidate at the 2010 general election.
Rochester and Strood: What's next for parties?
How unusual is Rochester and Strood?
Ms Thornberry is believed to have had two conversations with Labour leader Ed Miliband after posting the tweet, and offered her resignation during the second one.
In a statement released by the Labour Party, Ms Thornberry said: "Earlier today I sent a tweet which has caused offence to some people.
"That was never my intention and I have apologised. However I will not let anything distract from Labour's chance to win the coming general election."
"I have therefore tonight told Ed Miliband I will resign from the shadow cabinet."
Labour MPs said she had been right to stand down, Chris Bryant telling the BBC "the first rule of politics is surely that you respect the voters".
"She was absolutely wrong to tweet what she did," he said. "All I can say is, if somebody came into my constituency and did that, I would be furious."
And John Mann said the incident was "horrendous" for Labour.
"It insults people like me, it insults the people I know - my friends and family - Labour voters across the country because white vans, England flags, they're Labour values and actually pretty routine Labour values for most of us," he told Today.
He praised Mr Miliband's response, adding: "I think this is a different approach from the Labour leader and his message came out very, very clearly last night and she's had to go, she's been forced out."
But Mr Farage suggested the episode reflected broader attitudes within parts of the Labour Party.
"The Labour Party hate the concept of Englishness," he told the BBC News Channel. "They have done for a very long time.
"New Labour can't even stand the concept of patriotism. They think the flag somehow is unpleasant, backward-looking and nasty. People like Emily Thornberry would rather we had that blue flag with 12 stars on it that comes to us from Brussels."
Emily Thornberry has resigned from Labour's front bench after sending a tweet during the Rochester and Strood by-election which was branded "snobby".