Even if the possibilities in this highly touristic (and conservative) town are very tenuous, there are two spots for dumpster diving.
The first is outside of the covered market. Everyday, you can find vegetables around 1 pm. (Last checked: June 2012)
The second is the Rayons Verts Magasin Bio at 22 Rue Partouneaux, 06500 Menton. Sometimes, generally at 6h pm, you can find good organic vegetables or out-of-date vegan products. (Last checked: June 2012)
Another place for organic dumpster diving is "Naturalia" at 27 avenue Félix Faure. You can find food in front of the store after 8pm every day or after 2pm on Sundays. (Last checked: March 2018)
There's no dumpster diving possibility with supermarkets, they throw away the food directly to dustcart.
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