The announcement came just two before kick-off, leaving Leeds United against Arsenal as the only Premier League game to be fulfilled on Saturday.
This brings to six the number of Premier League that will be rescheduled this weekend.
A Premier League statement said: "After reviewing all the new information provided by Aston 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."
A club statement from Villa added: "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, 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."
