All animals with mobile eyes are equipped with a tearing system that keeps their eyes lubricated, protecting their eyes from the elements and so on. People tell tales of animals such as elephants and dogs weeping when punished. However, even if such an animal does shed tears, there is no scientific evidence that the animal is expressing emotion in doing so.
But what about when a baby chimp or ape whimpers and screams as it's being separated from its mother, couldn't we say with some certainty that the animal is crying? While this display does elicit caregiving from the mother, prompting her to feed or protect the baby, it is less clear that what the baby is feeling is similar to what a human feels when he or she cries.
In fact, it's not clear that crying and other emotional displays mean exactly the same thing to all humans. For instance, there is a wide range of emotions that prompt people to cry, and some bouts of weeping, such as crying at a stranger's wedding, seem to be culturally programmed responses.
In addition, autistics are said to be less emotionally expressive, but research has shown that it's more that their expressions and their reactions to the emotional expressions of others are unusual, different. But does that mean that their sadness is different somehow?
So even if apes can be said with some certainty to be feeling emotions, then there is the problem of figuring out to what extent what they feel resembles what humans feel.