It’s not stated in canon. MY personal inclination would be NO; they have crystalline eyes and no use for tear ducts. However, they are sustained by alchemical fluids, so if YOU want them to cry, you could explain it that way.

— Keith Baker (@HellcowKeith) December 17, 2019

But I’m inclined to focus on the differences between warforged and humans. They don’t express emotion as humans do. They can’t smile, so I wouldn’t have them cry; instead Id want to thinking of what they do to express sorrow INSTEAD of crying. If you’re looking for an involuntary response in place of crying, perhaps they have a slight tremor in their musculature that causes a faint chiming.

— Keith Baker (@HellcowKeith) December 17, 2019