The idea that entropy production puts a constraint on ecosystem functioning is quite popular in ecological thermodynamics. Yet, until now, such claims have received little quantitative verification. Here, we examine three 'entropy production' hypotheses that have been forwarded in the past. The first states that increased entropy production serves as a fingerprint of living systems. The other two hypotheses invoke stronger constraints. The state selection hypothesis states that when a system can attain multiple steady states, the stable state will show the highest entropy production rate. The gradient response principle requires that when the thermodynamic gradient increases, the system's new stable state should always be accompanied by a higher entropy production rate. We test these three hypotheses by applying them to a set of conventional food web models. Each time, we calculate the entropy production rate associated with the stable state of the ecosystem. This analysis shows that the first hypothesis holds for all the food webs tested: the living state shows always an increased entropy production over the abiotic state. In contrast, the state selection and gradient response hypotheses break down when the food web incorporates more than one trophic level, indicating that they are not generally valid.