"God alone is Lord of the conscience, and He has left it free from the doctrines and commandments of men which are contrary to His Word or not contained in it. Church and state should be separate. The state owes to every church protection and full freedom in the pursuit of its spiritual ends. In providing for such freedom no ecclesiastical group or denomination should be favored by the state more than others. Civil government being ordained of God, it is the duty of Christians to render loyal obedience thereto in all things not contrary to the revealed will of God. The church should not resort to the civil power to carry on its work. The gospel of Christ contemplates spiritual means alone for the pursuit of its ends. The state has no right to impose penalties for religious opinions of any kind. The state has no right to impose taxes for the support of any form of religion. A free church in a free state is the Christian ideal, and this implies the right of free and unhindered access to God on the part of all men, and the right to form and propagate opinions in the sphere of religion without interference by the civil power."
As a Presbyterian who holds to most of the Westminster Confession of Faith, and has become increasingly comfortable with the 1646 original version's section on the civil magistrate, I do hold some differences with this section. However, its broad principles and concerns for liberty of conscience are headed in the right direction: indeed God alone is Lord of the conscience, indeed the state may not interfere with the church's affairs in certain ways, and indeed the church's means of fulfilling its mission are spiritual only.
Men need not obey any doctrines or commandments of men--whether men of the state or men of the church--which are either contrary to the Word (civilly or in worship) or not contained in it (in worship); indeed men must disobey anything contrary to the Word.
Should church and state be "separate?" "Separate" is a slippery word, and it does not help that in common parlance today it is used somewhat interchangeably with other words like "distinct" and "different." In original meaning, though, "separation" is a much stronger idea than "distinction": "distinction" involves differences of constituent elements, qualities, or purposes; "separation" implies lack of close relationship. Scripture from beginning to end affirms a distinction between state and church as such.
Even in theocratic Israel under the Mosaic Law, a time during which the theocratic state of Israel was largely coextensive with the religious "assembly" of Yahweh, there were distinctions and limits on the duties of civic rulers versus the duties of priests. Indeed, except for rare instances (like David's dancing with the priestly ephod, foreshadowing Christ's king-priestliness), kings who dabbled in priestly matters were judged by God (cf. 1 Sam. 13).
In New Covenant terms, the state may not administer the two sacraments nor exercise church discipline. Likewise, the church may not carry out civic penalties like fines or corporal or capital punishment. Nevertheless, it is necessary and indeed unavoidable that these two different kinds of powers act in relation to each other, are coordinated with one another, and have reciprocal responsibilities in some ways. More on this below.
Does the state owe to "every church" protection and freedom to pursue its spiritual ends? It is my opinion that if "every church" includes "churches" or assemblies of any kind that publicly promote anti-catholic heresy or flagrant idolatry, a righteous state does note owe such assemblies anything but suppression (WCF 23.3). Likewise, when the BF&M says that no ecclesiastical group or denomination should be favored or supported by the state, I have to disagree if this means no favor for the Christian church over against false religious establishments like Islam or New Ageism. Where I might differ with classic Reformed establishmentarianism in my current thinking is that I do not think that 1646 Reformed Presbyterianism in all its confessional details (e.g. strict Lord's-Day civic Sabbatarianism) should be the law of the land; I believe in a broader Christian establishmentarianism, although I struggle with exactly where to draw those lines (i.e. should Roman Catholicism and Eastern Orthodoxy be treated as anti-Christian religions, or should such institutions be tolerated as severely errant yet sufficiently creedal Christianity?).
In any case, even a broad re-established Christendom would bring modern Western society much closer to the vision of the New Covenant prophetic ideal: "Thus says the Lord God, 'Behold, I will lift up My hand to the nations and set up My standard to the peoples; and they will bring your sons in their bosom, and your daughters will be carried on their shoulders. Kings will be your guardians, and their princesses your nurses. They will bow down to you with their faces to the earth and lick the dust of your feet; and you will know that I am the Lord; those who hopefully wait for Me will not be put to shame" (Is. 49:22-23).
The BF&M rightly states that Christians should be submissive to civil rulers short of rulers requiring disobedience to the Word of God. How this works out in practice can become complicated, especially once discussion turns to the issue of resistance by lesser magistrates, etc. We will not discuss those issues in detail here. The BF&M also rightly states that the church should not resort to civil power to carry out its work (i.e. it should not pick up the sword or threaten use of force to carry out the Great Commission of making disciples--that would be impossible and foolish, in any case). The church's weapons are spiritual, primarily: the proclaimed Word, sacraments, and prayer.
Does the state have "no right to impose penalties for religious opinions of any kind?" If these opinions are understood as opinions expressed and lived out in private, individually or within family households, it is correct. No legitimate biblical view of the establishment of God's Law in society--even the most continuous forms of theonomy--contemplates the civil magistrate breaking down doors on witch hunts for sexual perverts or idolaters. Civil law only touches public matters. There is also something practical to be said for the ability for different ideas to be publicly discussed and debated--even errant beliefs; how else would the gospel ever be able to be propagated among unbelievers? Nevertheless, it is the civil magistrate's duty to oppose flagrant public idolatry, blaspheming, and the unabashed promotion of sin and heresy. Such things are obviously "evil" against which the magistrate has been appointed by God to "bring wrath" (Rom. 13:3, 4). In other words, civil rulers, even in the execution of their office, must obey the 2nd Commandment (see WLC Q. 108).
This may sound strange to modern American ears, whose U. S. Constitution--at least as interpreted in modern times--prohibits any kind of establishment of religion, at any level of government, whatsoever. However, the modern liberal pluralist approach to religion and public life is incoherent, on close inspection. We are dealing with ultimate principles and absolutes, and therefore, if two different domains of power (church and state) swear allegiance--even implicitly--to different ultimate authorities (Jesus Christ vs. a sub-biblical notion of human "freedom" and the value of tolerance), and these two domains necessarily affect the acting out of values in the public sphere (which they both do), there is an inevitable clash and contradiction.
This becomes obvious when we think about how, in fact, on a liberal pluralist view of religious tolerance, the state ends up defining and restricting religion and its ethical boundaries, implicitly enforcing a religious claim. More concretely, in places like secular Canada where laws intended to promote religious "tolerance" and to avert "hate speech" have sometimes restricted public Christian evangelism, the enforcement of such laws cearly evince deeply religious values of a blatantly anti-Christian, supposedly "secular" state. Christianity requires public proselytizing. Christianity requires public opposition to the scourge of the Satanic ritual of legalized abortion. For the state to draw legal boundaries around the definitions and practices of religions like Christianity (or any other religion) is for the state to act as a religious authority. It is unavoidable. Therefore the answer is not to absolutize tolerance and secular, pluralistic "freedom" as a society's chief value; it is to promote the absolutization--through peaceful, spiritual weapons of gospel proclamation and discipleship (see a previous paragraph)--of true religion, namely, Christianity (to some degree of creedal specificity or another).
To put it as concisely as possible, in other words, there is no neutrality in civil government. There is Yahweh or no way. His truth, recorded for us in the Scriptures of the Old and New Testaments, is to be the absolute authority for lords, kings, presidents, congressmen, mayors, and judges. After all, Jesus Christ is King of kings and Lord of lords (not merely King of pastors and Lord of hearts). The expression of His authority is to become manifest in public society more and more, long before the full manifestation of His glorious kingdom on the Last Day when He returns bodily. God's Law, given robust expression under the Mosaic administration, with redemptive-historical adjustments we must continue to study and debate, must become the law of the land. There is no other consistent Christian view of public life.
The BF&M is right to say that a "free church in a free state is the Christian ideal." However, "freedom" must be defined consistently biblically as subjection to Christ and His Law both privately and publicly, which will lead to great blessing and even the "jealousy" of observing nations, as should have happened in Israel (Deut. 4:5-8; Jn. 8:34-36; Rom. 6:16-18; Jam. 2:12; Rev. 17:14; 21:24). To this end, the church is not to initiate violence. It is to disciple its own people, evangelize unbelievers, and prophetically critique ungodly laws and magistrates, until that spoken and written sword of the Spirit slays their wickedness and establishes godly laws and rulers in their place. This is a bottom-up, growing-mustard-seed approach to the coming of the kingdom of God on earth. "Thy kingdom come, Thy will be done on earth as it is in heaven," in homes, churches, and indeed in public offices and courtrooms.