Organisms that do not wish to reproduce
Animals and especially female animals might not "want" to reproduce when circumstances are bad for babies.  Pregnancy and rearing of infants is a big commitment of resources for female mammals.  If a female is under stress, or sick she might avoid having sex even though she could get pregnant because evolutionarily that was a winning strategy for her ancestors - committing the physical and nutritional resources to a pregnancy and fetus unlikely to survive current circumstances puts the mother and subsequent pregnancies at risk.  

A more extreme example is the Bruce Effect.  Pregnant females exposed to the pheromones of a strange new male "decide" not to reproduce - their bodies resorb the fetuses.  Presumably this is because the new male is a risk to the fetus and they would do better calling off the doomed project, and possibly having a new litter with the new male if he shows up.  

Weirder is the circumstance of parents that kill and sometimes eat their young.  That happens in mammals, birds, fish and others.  It is hard to see how this sort of thing advances genetic fitness - it seems like a crossed wire or misfire of some sort.  Certain animals, however, seem to do it all the time; hamsters come to mind.  Maybe stress causes weird behavior?  In any case this too would seem like a case of deciding not to reproduce.