Does Christianity approve wars for faith?

Pretty much, No.

In general, most Christians reject any idea of coercion in regards to "spreading the faith." Orthodoxy (from Orthos meaning right, and doxa meaning belief) is typically considered more important.

No less a luminary than Martin Luther, in commenting on Romans 12:1, for example, wrote:

“Paul does not say: I command you; for he is preaching to such as are already Christians and godly by faith, in newness of life. These must not be coerced by means of commandments, but admonished to do willingly what has to be done with the old sinful man in them. For any person who does not do this willingly, simply in answer to kind admonitions, is not a Christian; and any person who wants to achieve this result by force applied to such as are unwilling is not a Christian preacher or ruler, but a worldly jailer. A preacher of the Law comes down on men with threats and punishments; a preacher of divine grace coaxes and urges men by reminding them of the goodness and mercy which God has shown them. For He would have no unwilling workers nor heedless service; He wants men to be glad and cheerful in the service of God. Any person who will not permit himself to be coaxed and urged with sweet and pleasant words, which remind him of the mercy of God abundantly bestowed upon him in Christ, to do good joyfully and lovingly to the honor of God and for the benefit of his fellow men, is worthless, and all that is done for him is labor lost. If he is not melted and dissolved in the fire of heavenly love and grace, how can he be softened and made cheerful by laws and threats? It is not a man’s mercy, but the mercy of God that is bestowed on us; and this mercy Paul wants us to consider in order that we may be incited and moved by it to serve God.”

Historically, Just War Theory as promulgated by the famous fifth century theologian Augustine, said that a war should only be fought when protecting others from egregious harm - an extension of God's protection for the widow and orphan, as well an extension of the Golden Rule.

A good summary of jus ad bellum, meaning just causes for war are as follows:

Just Cause: War is permissible only to confront “a real and certain danger,” i.e. to protect innocent life, to preserve conditions necessary for decent human life existence, and to basic human rights.

Competent Authority: The right to use force must be joined with the common good; war must be declared by those with responsibility for public order, not by private groups or individuals.

Right Intention: War can be legitimately intended for only the reasons set forth as a just cause.

Last Resort: For war to be justified, all peaceful alternatives must have been exhausted.

Proportionality: The destruction to be inflicted and the costs incurred by war must be proportionate to the good expected by taking up arms. Destruction applies in both the temporal and spiritual sense.

Those incidents in which the faith was either "protected" or "extended" by coercion (say, the Inquisition, incidents during colonialization) are typically considered to be negative things by modern Christians today. Even the most right-wing and fundamentalist types like Bob Jones and Jerry Falwell never advocated for a return to these methods. Even Fred Phelps uses "Crusades" as a perjorative! Typically, the most common example of a "faith war" would be the Crusades, and to be sure, religion did play a part in those wars. Grant you,

• (a) that was nearly a thousand years ago and
• (b) the Fourth Crusade in particular, in which Christian forces from Italy sacked the Christian capital of Byzantium (1204), shows that there was a lot more involved than just religion,

but it is usually the argument. Most other "Christian" conflicts, like the Thirty Years War, fall into the same category - they were political conflicts as much as religious ones.