There had been cases of "verbal abuse hurled" at ethnic minorities, and "despicable graffiti daubed on a Polish community centre", he told the Commons.
Such attacks must be stamped out, he said, urging people to remember "these people have come here and made a wonderful contribution to our county".
Police are probing some incidents.
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Mr Cameron said "we have a fundamental responsibility to bring our country together" after the vote.
"And we will not stand for hate crime or these kinds of attacks. They must be stamped out," he said.
He said there would be "no immediate change" in the rights of EU nationals living in the UK, or UK citizens living in EU countries.
Since the vote, there have been reports of racism on social media, and a number of groups have reported a rise in incidents of racism and hate crime.
True Vision, a police-funded online reporting facility for hate crime, said it had received 85 reports between Thursday 23 and Sunday 26 June compared with 54 reports during the same period last month.
In London, the Metropolitan Police are investigating graffiti at a Polish community building in Hammersmith which they are treating as a hate crime "because of the racially-aggravated nature of the criminal damage".
Cambridgeshire Police are also investigating reports that laminated cards reading "Leave the EU - no more Polish vermin" were delivered to members of the Polish community in Huntingdon on Saturday.
Poland's ambassador to the UK Witold Sobkow said he was "shocked and deeply concerned by the recent incidents of xenophobic abuse directed against the Polish community and other UK residents of migrant heritage".
The full picture regarding hate crime after a Brexit vote could take months to emerge. And the national trend in recent years has been quite complicated.
Police-recorded hate crimes have been rising - but that could be more victims coming forward because they have confidence police will take them seriously.
The Crime Survey for England and Wales, an internationally-recognised rolling study of the bigger picture, suggests racism is slowly being pushed to the margins.
But there's no doubt that genuine moments of extreme national or global tension - such as major terrorist incidents or the conflict in the Middle East - are followed by spikes.
Extremists appear to take the opportunity to act because they think they can get away with it.
So the question is this: are the latest statistics a blip or a sign of a more worrying long-term trend?
Iman Etta, of Tell Mama UK, which collates reports of anti-Muslim abuse, said: "We've seen a spike in anti-Muslim hatred incidents and racist incidents reported in to us that are related to the result of the referendum, where perpetrators have been using words like 'we voted you out', 'you should be kicked out', and 'we no longer want you here'."
Muslim Remain campaigner Shazia Awan, a former Conservative parliamentary candidate, told the BBC's Victoria Derbyshire she had received several abusive tweets and emails.
One email read: "You are not now, nor will you ever be, Welsh. Being born in Wales has nothing to do with being Welsh.
"I cannot wait to send you and the anti-white garbage that you stand for back to the Third World dumps that you came from."
The Muslim Council of Britain said it had compiled more than 100 incidents of reported hate crimes since the result.
Dr Shuja Shafi, the body's secretary general, said: "Now we are witnessing the shocking extent of this with reports around the country of hate speech and minorities being targeted.
"Our country is experiencing a political crisis which, I fear threatens the social peace."
He said he would be writing to the home secretary to ask what measures were being taken to step up security and policing in areas where such incidences have been reported.
Mayor of London Sadiq Khan said he had asked the capital's Metropolitan Police to be on heightened alert for any rise in hate crime.
"I take seriously my responsibility to defend London's fantastic mix of diversity and tolerance," he said.
"So it's really important we stand guard against any rise in hate crimes or abuse by those who might use last week's referendum as cover to seek to divide us."
He added it was important not to demonise the 1.5 million people in London who voted for Brexit, and they should not be accused "of being xenophobic or racist".
Metropolitan Police Commissioner Sir Bernard Hogan-Howe said officers would "investigate vigorously any reports of crime motivated by hatred".
Karen Bradley, the minister for preventing abuse, exploitation and crime, said "nobody in this country should live in fear because of who they are".
"The government is working closely with communities to increase reporting and increase confidence that their concerns about hate crime will be taken seriously by the police and courts," she said.
No official statistics for reported incidents or racism or hate crime across the UK have yet been published. They are compiled by individual police forces.
The MP for Birmingham Yardley, Jess Phillips, has tweeted that she will ask a Parliamentary question on the issue.
However, numerous people have posted accounts of personal experiences on Twitter alongside the hashtag #PostRefRacism.
Sky News presenter Adam Boulton tweeted: "This weekend I and my family have witnessed 3 "when are you going home?" Racist incidents aimed at EU citizens here."
BBC journalist Sima Kotecha tweeted: "In utter shock: just been called p**i in my home town! Haven't heard that word here since the 80s..!"
Another user, James Titcombe, tweeted: "Daughter tells me someone wrote '[Child's name] go back to Romania' on the wall in the girls toilets at School today."
David Cameron has condemned "despicable" incidents of hate crime reported in the wake of the UK's referendum vote to leave the EU.