Is it all right to pierce our body when it is supposed to be the temple of the Holy Spirit? 

Although there is no scripture that directly prohibits the piercing of the body for decorative purposes, there are several teachings that should make believers beware of the practices that are common today. 

Leviticus 19:28Ye shall not make any cuttings in your flesh for the dead, nor print any marks upon you: I am the LORD.
See All... Ye shall not make any cuttings in your flesh for the dead, nor print any marks upon you: I am the LORD.

Leviticus 21:5They shall not make baldness upon their head, neither shall they shave off the corner of their beard, nor make any cuttings in their flesh.
See All... They shall not make baldness upon their head, neither shall they shave off the corner of their beard, nor make any cuttings in their flesh.

In these verses, the Lord prohibits the Israelites from making cuttings in their flesh as an act of mourning for the dead. Neither were they to print marks upon themselves. The Jews understood this last commandment to refer to tattoos. Tattooing was "a custom of the heathen, who used so to mark himself for idolatry, as much as to say that the tattooed person was a slave sold to the idol and marked for its service" (Mishneh Torah, Mada, Hilchoth Abodath Kochabim XII, 11). Although piercing the body for the sake of decoration is not being referred to here, piercing is in the same category as making marks on the body and cutting the flesh.

In Exodus 21:6Then his master shall bring him unto the judges; he shall also bring him to the door, or unto the door post; and his master shall bore his ear through with an aul; and he shall serve him for ever.
See All..., the servant who desires to be the servant of the master for life is to place his ear on the door or door post and "his master shall bore his ear through with an aul; and he shall serve him for ever." The boring, or piercing, of the ear is a sign of submission and slavery. One wonders what master the modern practitioner of body-piercing is submitting to.

When Elisha had his contest with the prophets of Baal on Mount Carmel, they "cut themselves after their manner with knives and lancets, till the blood gushed out upon them" (1 Kings 18:28And they cried aloud, and cut themselves after their manner with knives and lancets, till the blood gushed out upon them.
See All...). In Mark 5, Jesus met a man who "always, night and day, he was in the mountains, and in the tombs, crying, and cutting himself with stones" (Mark 5:5And always, night and day, he was in the mountains, and in the tombs, crying, and cutting himself with stones.
See All...). In most cases in the Bible, the cutting of the flesh is associated with idolatry or devil possession. Here are a couple of verses to consider:

1 Corinthians 10:23All things are lawful for me, but all things are not expedient: all things are lawful for me, but all things edify not.
See All... All things are lawful for me, but all things are not expedient: all things are lawful for me, but all things edify not.

1 Thessalonians 5:22Abstain from all appearance of evil.
See All... Abstain from all appearance of evil.

For the Christian, it is not just a matter of absolute right or wrong. It is also a matter as to whether or not it pleases the Lord. We should be motivated by love and desire to stay far away from those things that are displeasing to Him. We should avoid all appearance of evil and recognize that all things do not edify (or build up) the believer in His life for the Lord.

There is one more consideration for the piercing of the body. The Bible clearly warns against any extravagant decoration of the body for the purpose of bringing attention to it.

1 Timothy 2:9In like manner also, that women adorn themselves in modest apparel, with shamefacedness and sobriety; not with broided hair, or gold, or pearls, or costly array;
See All... In like manner also, that women adorn themselves in modest apparel, with shamefacedness and sobriety; not with broided hair, or gold, or pearls, or costly array;  10 But (which becometh women professing godliness) with good works.

1 Peter 3:3Whose adorning let it not be that outward adorning of plaiting the hair, and of wearing of gold, or of putting on of apparel;
See All... Whose adorning let it not be that outward adorning of plaiting the hair, and of wearing of gold, or of putting on of apparel;  4 But let it be the hidden man of the heart, in that which is not corruptible, even the ornament of a meek and quiet spirit, which is in the sight of God of great price.

This is not the place to discuss all of the teaching of these verses. And, admittedly, they do not directly speak of body-piercing. However, they do emphasize modesty in any kind of adorning for the body. Many of the practices we see today certainly go beyond modesty or moderation.

These biblical precepts have convinced me that body-piercing is something that every believer should approach with great caution. Clearly, a simple piercing of the ears is not the same as numerous piercings and ornaments all over the body. There is a strong element of personal conviction that tells us exactly where to draw the line. Yet, I have no doubt that the extent of piercing we see today is connected to true idolatry; people are serving the god of this world. It is not pleasing to the true God. As believers, we need to avoid going down this path. And, if we really love the Lord, we will see how far we can stay away from the line of disobedience, not how close we can get to it.