The tie will be played at the Etihad Stadium over the weekend commencing 26 January and ticket details will be released in due course.
This will be the fifth FA Cup meeting between the clubs, with City winning the last three clashes.
The Blues have already beaten the Clarets 4-1 at the Etihad this season.
The teams first met in 1931, with Burnley winning 3-0 at Turf Moor.
Two years later, the teams met again with City winning 1-0 in a tie also played at Turf Moor.
The next meeting was some 58 years later, again with the Blues drawn away, and again with City winning 1-0.
The full FA Cup 4th round draw is:
Arsenal v Manchester United
Crystal Palace v Tottenham
Barnet v Brentford
Portsmouth v QPR
Middlesbrough v Newport
Manchester City v Burnley
Chelsea v Sheffield Wednesday/Luton
Newcastle/Blackburn v Watford
Accrington Stanley v Derby or Southampton
Doncaster v Oldham
Brighton v West Brom
Bristol City v Bolton
Shrewsbury/Stoke v Wolves
Millwall v Everton
Swansea v Gillingham
AFC Wimbledon v West Ham