Why are some things said in the book of Leviticus still taught by the Church today but others aren't?
There are two kinds of precepts in the Old Testament. Leviticus gives us many examples of both. One kind of precept or law is a ceremonial or liturgical law regarding how worship is to be carried out and by whom. All of these laws of the Old Testament have been abrogated by the worship established by Christ in the New Testament—namely, the worship that consists of the Holy Mass and the sacraments and the order established by the apostolic authority of the Church.
The other kind of law in the Old Testament regards moral precepts. These, since they regard the moral law, which is substantially the same in the Old and the New Testament, are not abrogated but rather provide a kind of sense of what the minimum might be in the moral life, with Our Savior's Sermon on the Mount providing the maximum. The penalties and ways of judging crimes that are indicated no longer bind Christians, but the moral teachings do.
So, for example, having sexual relations with a person of one's own sex is immoral in the New and the Old Law, but in the New Law we are free to determine how we should deal with this sin socially. Thus, we no longer execute persons for this sin, nor for adultery, even though in the Old Testament even lesser sins might be so severely punished. Thus, we can see how in one sense the New Covenant makes stricter demands on the moral life of the individual while being in some cases less severe in the social consequences of sin. Our Lord's encounter with the woman taken in adultery illustrates this very well. "Go and sin no more," so the same moral law; but no stoning by other sinners, so a more merciful application.