Are Old Testament prohibitions against getting tattoos, mixing fabrics, and trimming sideburns required for Christians?
It's best to approach this subject with the understanding that the principles underlying all of the commandments, statutes, and judgments are universal and therefore are to be obeyed. The problem is that we don't know for sure what some of those laws that seem odd to us actually refer to. For example, the mixing of fabrics may have been forbidden because the priest's garments were of mixed fabrics, so to avoid confusion and keep the priesthood separate (the meaning of "holy"), the people did not wear garments of mixed fabrics. Or, it may be that merchants mixed cheaper fabrics with more expensive ones, producing inferior products while charging too much for them. The underlying universal principle, or law, is clear and is still in force today: Don't cheat folks! Whatever the case, it's certain that God doesn't have some kind of issue with what kinds of fabrics our clothing is made of, just as he doesn't insist on taking one's shoes off when in His presence (unless that's the custom showing respect and honor within a particular culture).
Tattoos, cuttings in the flesh, and cutting the edges of the hair and beard were apparently associated with pagan religious practices at the time, so the underlying law, which is still in place, is: Don't attempt to serve God the same way the heathen serve their false gods and stay far away from superstitious and magical rites.
It is probable that some of those practices were magical rites designed to bring good luck. God is simply telling the Israelites to rely on Him and avoid putting their trust in magic and superstition for healthy crops, babies, etc. Archaeologists have uncovered evidence for some of these customs, but there are still uncertainties about many of the specifics pertaining to them.
Because of my age and background, when I see tattoos, I do not associate them with anything positive. However, I realize that the people who get them today usually do so for reasons other than magical or religious or as a sign of servitude, and since I usually don't know their motives for getting tattooed, I cannot tell them they are in violation of God's law. The most I could do would be to say to them that I think their tattoos are unbecoming and that someday they will most likely regret having gotten them.
The specific commandments within the Torah tell us something about how the timeless, universal Law of God applies to specific circumstances common to the time in which the Law was given to Israel. For example, Moses commands that a parapet be put on the roof of one's house (Deuteronomy 22:8). The purpose of that law was for the protection of the people who would go up on the roof and use it as a living area. Have I dis- obeyed that law by not putting a parapet around the roof of my house? No, of course not. My roof is not used as a living area; I've never had or served a single meal upon it. But I am to obey the underlying universal principle, which means that if I dig a swimming pool in my backyard, I must fence it in or do something to make sure my neighbor's children don't wander into it. It's called loving one's neighbor as oneself—and that is the timeless principle (law) underlying the parapet instruction.
Here's another example: I recently underwent surgery—the surgeon cut into my flesh, and I have the scar to prove it. Now the law forbids cuttings in the flesh. So did I violate the law? No. The Bible does not forbid cutting the flesh for the purpose of fixing a problem. It does forbid cutting the flesh for the purpose of getting a "god" to do a favor for you (see 1 Kings 18:27–28).
In one way or another, all those laws that seem odd to us had some kind of purpose like that— either for the protection of human life or the preservation of God's way of life as opposed to heathen religious/magical practices.