Our 'muck heap' gets quite warm it steams away and no snow ever stays on it.
We had some damp hay once and we stacked it in the barn, a day later there was smoke issuing out of the barn and we had to hand-ball 100s of bales of damp (heavy), and very hot hay out into the yard - some of the bales actually ignited and started burning once the air/oxygen hit them.
A neighbouring farmer lost his barn and several other outbuildings and vehicles from storing damp hay - we were about 1/2 mile + away and realised what was happening when the barn roof (asbestos sheets) started exploding and then the sky lit up - a fire due to damp hay !
So, I'd say to you, damp material (vegetable matter, poop etc) WILL heat up.