Refrigerator fail due to adjacent chest freezer
Total, utter, male-bovine-derived-organic-fertilizer. 

The cabinet isn't helping it cool any (and is almost certainly less ventilation space than the manufacturer requires), but the freezer really can't have much effect, being on the other side of the cabinet. 

The stove on the other side gets far hotter than the outside of the freezer does - so why wasn't that blamed? Presumably because you added the chest freezer to an extant rental kitchen, and the landlord wants to stick you for price of the new refrigerator?

You are not, however, doing your chest freezer any favors if it was right next to the wood - most chest freezers have the coils running under the skin and also have a manufacturer-recommended minimum ventilation space around them.