It's very seldom used, I don't remember myself EVER using such sentence :)
If I see food on the table any time of the day and want to ask who it belongs to, I would say "Чиє це?" or "Чия це тарілка?" etc. This sentence does sound a bit unnatural to me.
If you see some set of the food on the table in the evening -- you can ask such question. Same could be applied to breakfast in the morning and lunch in the afternoon.
це is neuter, цей is masculine, and ця is for feminine. Correct? Вечеря is feminine. So why is it чия це вечеря rather than чия ця вечеря ?
Yes, це/цей/ця = n/m/f, corret. These pronouns mean "this" in an "adjective" sense, as in, they describe a noun: ця вечеря = this dinner.
Apart from that, це is a pronoun in a "noun" sense: it replaces a noun in a sentence. Це вечеря = this [is] dinner.
Чия це вечеря = Whose dinner is this?
Чия ця вечеря = Whose is this dinner?
Thanks very much sagitta145! That is a very clear reply, and that is exactly what I wanted to know.