You asked two related questions here.
- Would it assimilate dead organisms? (e.g - dead dogs)
- Would it assimilate non-living biomatter? (e.g. - food and compost)
Would it assimilate dead organisms?
The answer to the first question is no, the Thing is either incapable or uninterested in assimilating dead organisms.
In a deleted scene from the 2011 movie, Colin commits suicide by slicing his wrists rather than allow himself to be assimilated.
Although this a deleted scene and therefore non-canon, we see his body in the 1982 movie when the Americans visit the Norwegian camp, so his suicide is canon.
The Thing had a few days to assimilate Colin's body, but did not. From that we can assume it was either not interested or incapable.
Would it assimilate non-living biomatter?
The answer to your second question is unknown. To my best recollection, the Thing never assimilates (or just feeds on) non-living biomatter such as food or compost.
In this scene, the Thing destroyed the blood supplies rather than assimilating them. If it was capable of assimilating biomatter such as blood, it probably would have.
My hunch is that the Thing requires living cells because it requires biochemical metabolism to survive. Almost all cells with dead dogs and dead humans are not metabolically active, so the Thing ignores them. Nor are there any metabolically active cells within compost or food, so the Thing ignores them too.
Please note that the distinction is metabolically active rather than biochemically active. Not all biochemical reactions within cells allow the cell to maintain metabolism. Some reactions lead to cellular death, so even if dying/dead cells have biochemical reactions, the Thing's cells might not be able to infect them. Which is also why a cow-Thing can make milk that is safe for humans to drink.