It follows the Duke of Edinburgh's death, aged 99, on Friday morning.
However, hundreds of people had already congregated outside Buckingham Palace and Windsor Castle on Friday afternoon – though most appeared to be socially distanced.
While groups of up to six people, or two households, are now allowed to meet outdoors in England, lockdown laws ban large gatherings.
A Cabinet Office spokesman said: “Although this is an extraordinarily difficult time for many, we are asking the public not to gather at royal residences and continue to follow public health advice, particularly on avoiding meeting in large groups and on minimising travel.
Watch: Tributes paid to Prince Philip at Buckingham Palace
“We are supporting the royal household in asking that floral tributes should not be laid at royal residences at this time.”
Buckingham Palace also urged people not to gather in large numbers and suggested people make a donation to a charity rather than leave floral tributes.
In a statement, it said: "With the safety and wellbeing of the public in mind, and in accordance with government guidelines, members of the public are asked not to gather in crowds.
"Those wishing to express their condolences are asked to do so in the safest way possible, and not to gather at royal residences.
"During this time the royal family ask that members of the public consider making a donation to a charity instead of leaving floral tributes in memory of the Duke of Edinburgh."
Earlier on Friday, palace staff removed a framed plaque at the gate announcing Philip's death, in order to avoid a large congregation.
In the hour before it was removed, members of the public had been told to wear a mask and line up behind a barrier to view the sign.
Stewards later put a barrier around floral tributes placed at the gates of the palace in order to prevent overcrowding.
Members of the public were advised to join a queue to take pictures and place flowers, with stewards urging people not to gather in large crowds.
A Met Police spokeswoman also told Yahoo News UK that “the usual [COVID] rules apply” – though she added the force isn’t planning to issue a specific statement about people paying tribute to Philip.
In Windsor, mourners had earlier been chaperoned by police to lay flowers and share messages of support to the royal family.
Members of the public were later encouraged to leave any tributes at the Cambridge Gate on the Long Walk.
Well-wishers arriving with flowers at the Henry VIII Gate in the town centre were being directed to the other location by police officers.
Watch: The Duke of Edinburgh dies, aged 99