The clash at Villa Park becomes the latest match to be called off as clubs are hit by the COVID-19 omicron variant surges in the United Kingdom.
Villa manager Steven Gerrard said in a news conference on Friday that Premier League clubs will meet early next week to discuss the ongoing crisis.
Villa released a statement on Saturday that read: "Aston Villa can confirm that today's Premier League fixture with Burnley has been postponed due to an increased amount of positive COVID-19 test results within our playing squad.
"The results of PCR tests, which were taken yesterday prior to training as well as Lateral Flow Tests, were received this morning and confirmed further depletion of our playing squad which was already impacted upon for our trip to Norwich in midweek.
"All individuals who tested positive are now isolating in line with Premier League and government guidance and protocols.
"The club is extremely sorry for the inconvenience caused to Burnley Football Club, both our own supporters and the Burnley fans due to attend the fixture but have acted as swiftly as possible this morning to minimise disruption."
The match was scheduled to kickoff at 3 p.m. UK time but the postponement was announced just a few hours before.
Burnley chairman Alan Pace, who was Real Salt Lake's former CEO, tweeted: "Very disappointed like all of you that today's game has been postponed.
Very disappointed like all of you that today's game has been postponed. I know many of you, like myself will have made the journey to Villa and now face the long journey home. I ask all players, officials and spectators to stay safe. https://t.co/DuFQgz3iSq— Alan Pace (@AlanPaceBFC) December 18, 2021
"I know many of you, like myself will have made the journey to Villa and now face the long journey home. I ask all players, officials and spectators to stay safe."
It becomes the sixth match to be called off this weekend after Manchester United vs. Brighton, Southampton vs. Brentford, Watford vs. Crystal Palace, West Ham United vs. Norwich City, and Everton vs. Leicester were also postponed.
The Premier League also released a statement about the postponement which read: "After reviewing all the new information provided by Villa this morning and receiving separate medical advice, the Board approved the request as a result of the club not having enough players to field a team.
"This decision was based on the number of COVID-19 cases, injuries and illness. The League understands the decision to postpone the match at such short notice will frustrate and disappoint supporters, and apologises for the inconvenience and disruption caused.
"The remaining four Premier League matches due to be played this weekend are currently scheduled to proceed as planned.
"The board assesses applications to postpone matches on a case-by-case basis, based on existing rules and COVID-19 postponement guidance issued to all clubs."