"They are not your birds."
Translation:Vaši ptáci to nejsou.
This looks like the negative version of the familiar "to je/to jsou" construction, so I don't think you can leave out the "to." "Oni nejsou..." might be an option, but a Czech expert would be better to weigh in on that.
Actually, "Oni nejsou vaši ptáci." is fine, if somewhat unusual. It can sound like one is saying that about some people.