From what I read, not much (after 1941). Yet, it was a large force of men-up to 750,000 at its peak. It spent much of its time beseiging the city of Leningrad, and basically dug.
Also, it was commanded by a conservative, older general (Gen. von Leeb, who was probably sspect (of loyalty to Hitler).
Why was this vast force allowed to sit around, while the Army Groups Center and South were bleeding to death?
They were in fact doing the job they were assigned to do – in conjunction with the Finns pressing down along the Karelian Isthmus, to attempt to cause the surrender of Russia’s second largest city and former capital. As it was, they kept the Soviet manpower defending Leningrad from being brought to bear against the other Army Groups.
Plus strategically, this would immediately cut Russia off from the Baltic, and would open the way to cut the lines supplying Russia from Murmansk and Arkhangelsk.
I heard somewhere they originally did want to take Leningrad by force, but Hitler wanted them to lay siege instead. It was supposedly easier in the long run to have everyone starve rather than take the city and have to exterminate/enslave/deport them all later.
No cite, but it would be typical to siphon troops away from one Army Group to others which are having trouble or which are considered to have more vital missions. I would expect that AG North was plundered for mobile units (especially tanks) to reinforce the drives on Moscow and later Stalingrad, leaving an understrength AGN to try to strangle Leningrad.
Hrm, quick Googling produces some support for this possibility:
http://www.gutenberg-e.org/esk01/frames/fesk03.html
Note that the sequence of events described above resulted in German front-line mobile formations leaving AGN and later being replaced by mines and an allied Spanish infantry division, presumably of lower effectiveness.
Dat’s a good question.