Why is Mary's perpetual virginity a Catholic dogma?

If she had relations with Joseph at some point after their marriage, what truth of or about Christianity would be damaged?

She would've sinned by violating her vow of virginity. However, she is sinless.

The prophecies (e.g., Isaiah's) that the Christ would be born of a virgin wouldn't have been fulfilled.

How is Christianity diminished if she did not remain a virgin?

St. Thomas Aquinas, answering the question of whether Christ should have been born of a virgin, gives several reasons (Summa Theologica III q. 29 a. 1):

It was fitting that Christ should be born of an espoused virgin

1. for His own sake
   1. lest He should be rejected by unbelievers as illegitimate: wherefore Ambrose says on Lk. 1:26,27: "How could we blame Herod or the Jews if they seem to persecute one who was born of adultery?"
   2. in order that in the customary way His genealogy might be traced through the male line. Thus Ambrose says on Lk. 3:23: "He Who came into the world, according to the custom of the world had to be enrolled Now for this purpose, it is the men that are required, because they represent the family in the senate and other courts. The custom of the Scriptures, too, shows that the ancestry of the men is always traced out."
   3. for the safety of the new-born Child: lest the devil should plot serious hurt against Him. Hence Ignatius says that she was espoused "that the manner of His Birth might be hidden from the devil."
   4. that He might be fostered by Joseph: who is therefore called His "father," as bread-winner.
2. for His Mother's sake
   1. because thus she was rendered exempt from punishment; that is, "lest she should be stoned by the Jews as an adulteress," as Jerome says.
   2. that thus she might be safeguarded from ill fame. Whence Ambrose says on Lk. 1:26,27: "She was espoused lest she be wounded by the ill-fame of violated virginity, in whom the pregnant womb would betoken corruption."
   3. that, as Jerome says, Joseph might administer to her wants.
3. for our sake.
   1. because Joseph is thus a witness to Christ's being born of a virgin. Wherefore Ambrose says: "Her husband is the more trustworthy witness of her purity, in that he would deplore the dishonor, and avenge the disgrace, were it not that he acknowledged the mystery."
   2. because thereby the very words of the Virgin are rendered more credible by which she asserted her virginity. Thus Ambrose says: "Belief in Mary's words is strengthened, the motive for a lie is removed. If she had not been espoused when pregnant, she would seem to have wished to hide her sin by a lie: being espoused, she had no motive for lying, since a woman's pregnancy is the reward of marriage and gives grace to the nuptial bond." These two reasons add strength to our faith.
   3. that all excuse be removed from those virgins who, through want of caution, fall into dishonor. Hence Ambrose says: "It was not becoming that virgins should expose themselves to evil report, and cover themselves with the excuse that the Mother of the Lord had also been oppressed by ill-fame."
   4. because by this the universal Church is typified, which is a virgin and yet is espoused to one Man, Christ, as Augustine says (De Sanct. Virg. xii).
   5. since the Mother of the Lord being both espoused and a virgin, both virginity and wedlock are honored in her person, in contradiction to those heretics who disparaged one or the other.

If she had relations with Joseph at some point after their marriage, what truth of or about Christianity would be damaged?

Helvidius believed that Mary knew Joseph carnally after the birth of Jesus and that she bore other children. Helvidius's error is

1. derogatory to Christ's perfection: for as He is in His Godhead the Only-Begotten of the Father, being thus His Son in every respect perfect, so it was becoming that He should be the Only-begotten son of His Mother, as being her perfect offspring.
2. an insult to the Holy Ghost, whose "shrine" (sacrarium) was the virginal womb, wherein He had formed the flesh of Christ: wherefore it was unbecoming that it should be desecrated by intercourse with man.
3. derogatory to the dignity and holiness of God's Mother: for thus she would seem to be most ungrateful, were she not content with such a Son; and were she, of her own accord, by carnal intercourse to forfeit that virginity which had been miraculously preserved in her.
4. would be tantamount to an imputation of extreme presumption in Joseph, to assume that he attempted to violate her whom by the angel's revelation he knew to have conceived by the Holy Ghost.

^{—St. Thomas Aquinas, Summa Theologica III q. 28 a. 3 "Whether Christ's Mother remained a virgin after His birth?" c.}

Why is it dogma? For the same reason that the Nicene-Constantinople Creed was established and made official.

It is an explicit statement of what the Church believes (as seen from the point of view of the Church, which speaks for the Faithful). Pope Pius IX felt that it was important to spell this out. Some beliefs held by the faithful, on the unofficial level, had not been confirmed officially for centuries. In this case some were in conflict so a resolution was provided, and was grounded in extant Catholic theology, belief, and tradition.

Previous to the proclamation of the Dogma, some Marian traditions and beliefs fell into Opinio Tolerata.

Before 1854, when Pope Pius IX defined the dogma of the Immaculate Conception in his papal bull Ineffabilis Deus, some Catholics were of the opinion that Mary was conceived without sin, while others doubted this privilege as being given to Mary. Once this decree was promulgated on December 8, 1854, Catholics must believe that the Immaculate Conception is dogma.

In layman's terms, the Magisterium finally stopped procrastinating since there were diverging traditions. This answer goes into some detail about various Marian beliefs (which go back to the third century) taking more formal shape and being codified, but the motivation was spelled out in Ineffabilis Deus. @Geremia presented this in answer to a different question here, extracted from ID(extracted from ID).

Supreme Reason for the Privilege: The Divine Maternity

And indeed it was wholly fitting that so wonderful a mother should be ever resplendent with the glory of most sublime holiness and so completely free from all taint of original sin that she would triumph utterly over the ancient serpent. To her did the Father will to give his only-begotten Son -- the Son whom, equal to the Father and begotten by him, the Father loves from his heart -- and to give this Son in such a way that he would be the one and the same common Son of God the Father and of the Blessed Virgin Mary. It was she whom the Son himself chose to make his Mother and it was from her that the Holy Spirit willed and brought it about that he should be conceived and born from whom he himself proceeds.

... additional reasons:

…for the honor of the Holy and undivided Trinity, for the glory and adornment of the Virgin Mother of God, for the exaltation of the Catholic Faith, and for the furtherance of the Catholic religion

This is consistent with what St Thomas Aquinas presented in the Summa Theologica, regarding Whether Christ's Mother remained a virgin after His birth?:

... It is written (Ezech. 44:2): "This gate shall be shut, it shall not be opened, and no man shall pass through it; because the Lord the God of Israel hath entered in by it." Expounding these words, Augustine says in a sermon (De Annunt. Dom. iii): "What means this closed gate in the House of the Lord, except that Mary is to be ever inviolate? What does it mean that 'no man shall pass through it,' save that Joseph shall not know her? And what is this---'The Lord alone enters in and goeth out by it'---except that the Holy Ghost shall impregnate her, and that the Lord of angels shall be born of her? And what means this---'it shall be shut for evermore'---but that Mary is a virgin before His Birth, a virgin in His Birth, and a virgin after His Birth?" ... we must abhor the error of Helvidius, who dared to assert that Christ's Mother, after His Birth, was carnally known by Joseph, and bore other children. For, in the first place, this is derogatory to Christ's perfection: for as He is in His Godhead the Only-Begotten of the Father, being thus His Son in every respect perfect, so it was becoming that He should be the Only-begotten son of His Mother, as being her perfect offspring. (Note: Aquinas goes on to raise three other objections, see the link).

Credit is due to @Geremia for providing the link that addresses the full objection to Helvidius' position, and further substantiation for the belief that Mary remained virgin after the birth of Jesus.

A summary of comments on this topic from other theologians and doctors of the Church can be found here, as well as commentary on the Protoevangelium of James (circa 120 AD) which takes the same position.