The game was scheduled to be played at the Vitality Stadium on Saturday, but will now take place on Tuesday 26 January with a 7pm kick-off, with Crawley's League Two clash against Walsall pushed back until March 16 as a result.
Town, who stunned Leeds in the third round, were forced to suspend Tuesday night's league fixture against Stevenage after players and first-team staff were required to self-isolate.
An FA statement read: "Crawley Town's fixture against AFC Bournemouth in the Emirates FA Cup fourth round has been rescheduled following a decision by our Professional Game Board.
"The fixture is now scheduled to take place at the Vitality Stadium on Tuesday 26 January at 7pm and will be broadcast live in the UK on BT Sport Extra 3.
"Crawley informed us that there have been a significant number of positive Covid-19 cases at the club and a number of further players are currently self-isolating.
"We requested and considered further information regarding the club's ability to fulfil the fixture, which included discussions with our Covid-19 medical officer for the Emirates FA Cup and details of the medical advice that the club has received, which was used by the Professional Game Board to make its decision.
"We would like to thank the EFL, Crawley Town, AFC Bournemouth, Walsall and BT Sport for helping to make the rescheduling of this fixture possible and wish full and speedy recoveries to all those who have tested positive for Covid-19."