Can the Thing infect non-living biomatter?

You asked two related questions here.

1. Would it assimilate dead organisms? (e.g - dead dogs)
2. 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.

Summary

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.

There are no indications that the Thing can assimilate anything other than living cellular matter.

When Macready and Copper visit the Swedish camp, they find that one of people there has apparently slit his own wrists; presumably he, like Fuchs later on, killed himself rather than be assimilated. However, while Fuchs apparently immolated himself, to render his body useless to the Thing, the corpse of the other suicide was left alone to freeze with the destruction of the camp—suggesting that the surviving dog Thing present at the camp was not able to assimilate the dead matter.

There are no other indications in the 1982 film that the Thing can make use of any other nonliving matter, even if that matter is of organic origin. It does not assimilate the blood, for example, and we know that the Thing can mimic human blood, as Palmer shows. Nor does it seem to have invaded the outpost’s food supplies. Whether living plants can be infected is a separate and unaddressed question, since the scene with the pot farm was not included in the final film.

Finally, it is quite clear in “Who Goes There,” the original story on which The Thing was based, that nonliving organic matter is not dangerous. In addition to the dogs, the (much larger) Antarctic base in the story has a herd of milk cows, and some time passes before anyone realizes that all the animals have been taken over. However, the milk produced by the assimilated cows (which the men have been drinking) is quite harmless.

McReady shrugged his shoulders hopelessly. He went over to the milk bucket, and with his little tube of serum went to work on it. The milk clouded it, making certainty difficult. Finally he dropped the test­tube in the stand and shook his head. "It tests negatively. Which means either they were cows then, or that, being perfect imitations, they gave perfectly good milk."

Since the milk is only an hour old at this point (and the guy who went to milk them turned out to be Thing himself), the cows were almost certainly infected when the milk was collected. Had it been possible to assimilate the milk, it would certainly have made sense for the Thing to have done so—thus enabling it to take over anyone who drank it from inside.