It's hit or miss.
You need to make sure that your VVS airbases have decent level bombers set to night missions to supply the partisans. In 41, I use DB, TB and SBs. And I tend to keep using these due to the long range. Otherwise, very little to nothing will happen.
Flav gave me a good suggestion a month or so back - put three VVS airbases under three BAK HQs, split the front up into thirds to cover the area to supply the partisans, put one HQ group (HQ, 3 ABs) to cover each third, and keep them close enough to do their job and far enough back not to get over run. Then check them occasionally to make sure the air units stay up to strength. Works well and doesn't take much time.
From what I can tell, partisans will only break rail lines. And while their numbers go up as the years go on, the influence actually drops a lot after the first blizzard. By then, the German RR units have connected a lot of the cross RR lines in the back so a single attack won't cut supply lines. You would have to have two or three in an area to have more than a minor impact.
And per some other post, partisans are like other units - you have to reach an effectiveness score for TOE being filled out + training (or morale I can't remember) for a unit to attack a rail line. The attack shows up in the phasing between player turns when supply, leader change out, etc.. happens.
In 41, a single partisan, given some luck and at the right time, can put a lot of Germans in a supply bind. And if the German player isn't paying attention, he may even miss the reason for the lack of supply. Other than that, I haven't seen any impact to speak of in the game.
After the first blizzard, I haven't just don't see much impact. But then again, they don't really take up much time or resources.
Would be interested to hear what others have seen.