The Thai embassy's consular section in the Burmese capital said it would be closed for the week in the wake of the "unexpected and prolonged" protests.
Win Zaw Htun and Zaw Lin were sentenced to death last week for the 2014 murders of David Miller and Hannah Witheridge.
Thai police have denied allegations the men were forced to confess.
But many in Myanmar believe the men were made scapegoats for the crime. The convicted migrant workers denied killing the tourists, saying they were forced by Thai police into confessing under duress, which they later retracted.
Hundreds of people, including Buddhist monks and ordinary citizens, were reported to have taken to the streets on Tuesday. Thai media has earlier reported that Thai authorities asked Myanmar to contain the protests.
A statement made by the Thai embassy released earlier this week said the demonstrations made it hard to access their entrance.
After the verdict hundreds took part in protests which erupted outside the Thai embassy in Yangon, as well as in some towns across Thailand, after Thursday's court ruling, calling for the men's release.
Myanmar's army chief Gen. Min Aung Hlaing has called for the case to be reviewed, to which Thailand's Prime Minister Gen. Prayuth Chan-ocha said that the men had the standard right to appeal, according to local media.
Hannah Witheridge, 23, from Norfolk, and David Miller, 24, from Jersey, were found dead on the island of Koh Tao on 15 September last year.
The family of Mr Miller say they believe justice was done by the court, in which "the judges were invariably diligent, attentive, fair and extraordinarily hardworking," said David's brother, Michael.
Thai police spokesperson Gen Dejnarong Suthichanbancha  has consistently defended the investigation saying it "was done transparently and  in compliance with international standards".
Protests against the conviction in Thailand of two Burmese men for the murder of a pair of British tourists have been growing in Yangon.