First, you need to seal those doors so you're not constantly cleaning up the floor! I am surprised that this problem is as 'large' as you make it. I mean, did someone cut the trees for you and make the doors? Hearing this from a store-bought door is a bit unusual. Granted, there may be some bleeding but dripping??!
Anyways, try squirting some WD-40 on to a rag and gently rubbing the sap in the carpet between your fingers. Kinda like you're making a ball out of putty. The WD-40 in the rag should soften up the sap. If this doesn't work, get a new rag and wet it with denatured alcohol, and do the same thing with the 'ball rubbing'. And if this doesn't work, try Mineral Spirits. Just work the area/spot w/the sap in it. DON'T go crazy and start wiping the rag all over the carpet!