Burnley fans are hugely loyal by nature.
I've watched The Clarets play for over 50 years so I speak from experience when I say that any call for a manager's head from the stands would be classed as a collector's item.
One of the first songs I ever sung in the early 1970's was 'Bob Lord out!' but crucially he was a chairman - when we turn at Turf Moor we tend to go for the owners, not the man in the dugout.
So I would say (fairly) categorically that despite Burnley being second from bottom of the Premier League, Vincent Kompany has nothing to worry about at the moment as far as supporters are concerned.