The idea, believe it or not, comes from Scripture. The most compelling clue comes from the story of Nadab and Abihu. These priests, sons of Aaron, offered "strange fire" and were instantly burned to death (Lev 10:1-2). For "strange fire"? Oh, sure, the ESV says "unauthorized fire", like that helps. The point is that they didn't violate a command from God; they simply did something in worship that He had not commanded. When Aaron started to complain, Moses told him, "This is what the LORD has said: 'Among those who are near Me I will be sanctified, and before all the people I will be glorified.'" And the text says that with that, "Aaron held his peace." (Lev 10:3) It looks then like God is concerned with specifics in the worship He receives. Thus, Moses was not allowed to make whatever he thought appropriate for the tabernacle. He had to make everything "after the pattern for them, which is being shown you on the mountain." (Exo 25:40) The first two commands of the Decalogue are about the proper worship of God (Exo 20:1-6). Paul warns about "self-made religion" which "have indeed an appearance of wisdom" but "are of no value in stopping the indulgence of the flesh." (Col 2:23) Jesus rejected "the traditions of the elders" (Matt 15:1-14) and required a return to "the commandment of God". So it looks like the regulative principle of worship may have a biblical case.
After that, of course, the case breaks down. No, I don't mean it goes away. I mean it gets difficult to interpret. If we are going to accept that the worship of God must only be that which God commands, what does God command? Well, there is the reading of Scripture (1 Tim 4:13) and preaching the Word (2 Tim 4:2). There is singing (Eph 5:19; Col 3:16) and prayer (Matt 21:13). There is baptism (Matt 28:19) and the Lord's Supper (1 Cor 11:23-26). These are pretty agreeable. And then the followers of the regulative principle of worship diverge.
How often do we celebrate the Lord's Supper? What about dancing? Drama? Videos? Technology at all? What about instruments? If instruments, what instruments? What songs are we allowed to sing? Many require that all singing for the purposes of worship be done solely from Scripture. Lots of unanswered questions. The Reformed Presbyterian Church of North America holds to the most extreme version of this principle that I know of. From their website:
We believe that God’s Word clearly sets forth how He is to be worshiped. The reading and exposition of the Word of God are the central focus of our worship. Our musical praise employs God’s Word only, thus making use of the divinely inspired Book of Psalms of the Bible. In keeping with the New Testament Church’s directive for heart worship, we sing without the aid of musical instruments.Only God's Word is sung with only the Psalms and without musical instruments. Now, Paul commands us to "Let the word of Christ dwell in you richly, teaching and admonishing one another in all wisdom, singing psalms and hymns and spiritual songs, with thankfulness in your hearts to God." (Col 3:16) Doesn't that mean it's more than just the Psalms? And Psalm 150 says to praise Him with instruments. Doesn't that mean that instruments are not a problem? "Oh," some will say, "we can use instruments, but only those listed in Scripture."
I think the case for the regulative principle of worship has merit. Worshiping God apart from what God wants wouldn't make sense at all. And I'm pretty sure that many Christians today are quite cavalier about their worship to God. They base it on "a warm feeling toward God" when worship is about God, not our feelings. A cool band with contemporary singing may feel good to you, but does it reach God? On the other hand, figuring out exactly what that means -- what is and isn't commanded -- seems to be problematic for us. At this point don't we end up back with the "Christian Liberty" thing? Oh, I don't know.