Does the Bible say anything specific about flirting?
In general, no the Bible does not say anything about flirting. There is some flirting in the Song of Songs (Song of Solomon), which some have said is about God's relationship with His people and others have said was about courtship and marriage. That flirting, though, was apparently between a woman and her husband.
If one party to the flirting is married (or both parties, but not married to each other), then the Bible may have something to say about that. Genesis 2:24 (quoted in Matthew 19:5 and Mark 10:7) indicates that marriage puts all others "off limits." Because of this, if a spouse objects to the other flirting with anyone else, that would make it wrong. It may even be considered wrong if the flirting goes on even with the other spouse's approval. Since flirting is a form of sexual play, flirting with someone other than your spouse could be considered wrong.
If you aren't married, is flirting wrong? The Bible doesn't say. It is a normal part of getting to know a potential mate. If it goes beyond flirting and into a sexual encounter outside of marriage, then it becomes wrong. But I find nothing that the Bible says about flirting itself.