Piper's article refers to an item by Greg Koukl, actually a transcript of a radio commentary on the subject of contraception, and it's titled "Birth Control and God's Will".
Koukl was responding to a caller who said, "If you are practicing birth control then you are thwarting God's sovereign will." That's a ridiculous notion, because God's sovereign will cannot be thwarted. Otherwise it wouldn't be sovereign.
Assuming the caller meant that God's moral will, is the use of birth control a violation of God's moral will.
Koukl argues from the perspicuity of Scripture. Perspicuity is a foundational principle of the Reformation. Chapter I of the Westminster Confession of Faith, Of the Holy Scripture, states:
All things in Scripture are not alike plain in themselves, nor alike clear unto all: yet those things which are necessary to be known, believed, and observed for salvation are so clearly propounded, and opened in some place of Scripture or other, that not only the learned, but the unlearned, in a due use of the ordinary means, may attain unto a sufficient understanding of them.
As proof of that principle, the WCF cites Psalm 119:105, "Thy word is a lamp unto my feet, and a light unto my path," and Psalm 119:130 "The entrance of thy words giveth light; it giveth understanding unto the simple."
If we don't fulfill His moral will, if we violate His moral will, then we are in sin so God tells us what is wrong in order that we might be careful not to do what is wrong and instead do what is right. This is why when it comes to moral will, the moral will of God is clearly articulated throughout the Scriptures. It is all the things we ought to do and we ought not do, and they deal not just with our actions but with our attitudes and our motives. It is all inclusive.
Asking whether Onan's fatal transgression in Genesis 38 was birth control by means of coitus interruptus, Koukl says:
The question occurs to me, if God feels so strongly about such a thing, why are we left up to our own devices to figure this out from one verse in the book of Genesis? Why doesn't He confirm such a hatred for this act in the Law? You've got chapters and chapters and chapters in Exodus and Leviticus and Deuteronomy of God's feelings about the conduct of His people. We have dozens of references to sexual behavior, prohibitions and concerns, such that God's people can be careful to live and act properly in their sexual and procreative life. Yet not a single mention is made of coitus interruptus.
We have the prophets speaking volumes of God's intentions about the conduct of His people, and no mention of such a thing in the prophets. We have the New Testament to clarify anything that was confusing in the Old Testament. Jesus gave His teaching, then the Apostles, Peter and James and John and Paul--volumes of information clarifying the law so that we might not be confused about proper conduct. Where is the mention against birth control--or even one form, coitus interruptus? There are none. There are no direct or even mildly indirect statements about such a thing.
As I'll get to in a later entry, at least one evangelical pastor believes that God's act against Onan in Genesis 38 is a sufficiently clear statement from God.
But it seems to be a common principle of evangelical hermeneutics that we have to be very careful about deducing universal commands or obligations from the Bible's historical narrative. E.g., we shouldn't assume because God rewarded Jabez for his prayer that we'll receive the same reward if we pray exactly the same prayer.
The Catholics get around this issue by deriving their opposition to birth control from natural law, which they then use as a grid for understanding the Scripture.
What you have to be careful about is "proving too much." Supposing I have an interpretive principle that leads me to infer a prohibition on all contraception. What are the consequences if I apply that principle to other cases? What other commands which are not explicitly declared in Scripture can I declare as holy obligations for all true Christians?
The Catholic natural law approach opens the door for all sorts of Pharisaical burdens to be placed on the faithful. (Evangelicals, particularly the more fundamental variety, are plenty good themselves at creating extrabiblical "commandments" and reasons for feeling false guilt, although they use a different approach to get there.)