A Lewis McGugan penalty in injury time saved the day for Forest after Kevin Long had collided with Henri Lansbury in the box.
Junior Stanislas' second-half goal had looked to be enough for the Clarets in a contest between two tired teams.
The busy Easter fixture list had forced Davies to make six changes from the side that snatched a last-minute point against Brighton on Saturday.
Davies told the club's website: "I have to apologise to our fans for having to watch that type of football. It wasn't a good game - there's no doubt that we didn't play at the tempo or quality we expect of our players.
"Having said that, Burnley had an extra 24 hours and we were forced into making changes because of that. Sam Hutchinson wasn't in good condition after playing his first game in six months against Brighton and Adlene Guedioura had concussion.
"We also tried to freshen up the two strikers because two games in three days is very difficult. All six were forced changes because of the small gap between the two matches.
"I felt quite confident that we would get something, even at the end. This team never gives in, as was proved again today.
"Turf Moor is a very difficult place to come and it's pleasing to go away with a point."