Focusing strictly on the question of the Biblical basis for believing that breaking the civil laws of the State is sinful, here are some of the Bible passages implying or stating that we should obey the civil authorities (which, in the case of the ancient Jewish State, were the same as the religious authorities).

These are quoted in their Biblical order.

First from the Gospels:

When Jesus had come down from the mountain, great crowds followed him; and there was a leper who came to him and knelt before him, saying, "Lord, if you choose, you can make me clean."

He stretched out his hand and touched him, saying, "I do choose. Be made clean!" Immediately his leprosy was cleansed. Then Jesus said to him, "See that you say nothing to anyone; but go, show yourself to the priest, and offer the gift that Moses commanded, as a testimony to them." (Matthew 8:1-4; see also the parallel passages in Mark 1:40-45; Luke 5:12-16)

Here Jesus instructs a man he has healed to go to the priest and offer the gift commanded in the Law of Moses, "as a testimony to them." This begins to establish a theme of obeying the authorities as a way of testifying to the power and sovereignty of God.

The next passage provides a different angle on this theme:

When they reached Capernaum, the collectors of the temple tax came to Peter and said, "Does your teacher not pay the temple tax?"

He said, "Yes, he does."

And when he came home, Jesus spoke of it first, asking, "What do you think, Simon? From whom do kings of the earth take toll or tribute? From their children or from others?"

Peter said, "From others."

Jesus said to him, "Then the children are free. However, so that we do not give offense to them, go to the sea and cast a hook; take the first fish that comes up; and when you open its mouth, you will find a coin; take that and give it to them for you and me." (Matthew 17:24-27)

Here Jesus argues that "the children," meaning citizens of the nation, are free from the taxes imposed by the authorities—in this case, the religious authorities. Yet he tells Peter to pay the temple tax anyway, "so that we do not give offense to them." This is commonly read as establishing a theme of obeying civil law so as not to offend against the civil authorities and the sensibilities of the people, who hold the civil law in esteem for keeping civil order and punishing evildoers.

This theme is developed more fully in 1 Peter 2:11-17, quoted in its Biblical order below.

Then the Pharisees went and plotted to entrap him in what he said. So they sent their disciples to him, along with the Herodians, saying, "Teacher, we know that you are sincere, and teach the way of God in accordance with truth, and show deference to no one; for you do not regard people with partiality. Tell us, then, what you think. Is it lawful to pay taxes to the emperor, or not?"

But Jesus, aware of their malice, said, "Why are you putting me to the test, you hypocrites? Show me the coin used for the tax." And they brought him a denarius. Then he said to them, "Whose head is this, and whose title?"

They answered, "The emperor's."

Then he said to them, "Give therefore to the emperor the things that are the emperor's, and to God the things that are God's." When they heard this, they were amazed; and they left him and went away. (Matthew 22:15-22; see also the parallel passages in Luke 20:20-26; Mark 12:13-17)

Though the message here is nuanced, it is commonly interpreted to mean that as long as we are under a civil authority, we should pay the taxes required by the civil authority, and in broader terms, obey the civil authorities in matters in which it has jurisdiction, while also obeying the law of God where it has jurisdiction (which would presumably be in all significant matters of conduct).

Then Jesus said to the crowds and to his disciples, "The scribes and the Pharisees sit on Moses' seat; therefore, do whatever they teach you and follow it." (Matthew 23:1-3)

Here Jesus tells his hearers to obey the currently sitting religious authorities, even though in other places he commonly refutes their interpretation of the law and states that they have nullified the law by their human edicts.

And from the Epistles:

Let every person be subject to the governing authorities; for there is no authority except from God, and those authorities that exist have been instituted by God. Therefore whoever resists authority resists what God has appointed, and those who resist will incur judgment. For rulers are not a terror to good conduct, but to bad. Do you wish to have no fear of the authority? Then do what is good, and you will receive its approval; for it is God’s servant for your good. But if you do what is wrong, you should be afraid, for the authority does not bear the sword in vain! It is the servant of God to execute wrath on the wrongdoer. Therefore one must be subject, not only because of wrath but also because of conscience. (Romans 13:1-5)

This one the clearest and most extensive Biblical injunctions to obey the civil authorities.

Its basis is different than that of many of the other passages quoted in this answer. Rather than stating that we should obey the civil authorities so as not to give offense, and so as to be examples of good and law-abiding citizens, thus drawing people to God, this passage states that God has established the civil authorities in their positions, and that obeying them is obeying God's agents. Failing to do so will bring judgment down upon the heads of the lawbreakers.

This could be interpreted as incurring civil judgment and penalties, or as incurring divine judgment—which would imply that breaking the civil law is sinful, and will be subject to God's judgment.

A brief but on-point statement:

Remind them to be subject to rulers and authorities, to be obedient, to be ready for every good work. (Titus 3:1)

Another clear and extensive Biblical injunction to accept and obey civil authorities:

Beloved, I urge you as aliens and exiles to abstain from the desires of the flesh that wage war against the soul. Conduct yourselves honorably among the Gentiles, so that, though they malign you as evildoers, they may see your honorable deeds and glorify God when he comes to judge.

For the Lord's sake accept the authority of every human institution, whether of the emperor as supreme, or of governors, as sent by him to punish those who do wrong and to praise those who do right. For it is God's will that by doing right you should silence the ignorance of the foolish. As servants of God, live as free people, yet do not use your freedom as a pretext for evil. Honor everyone. Love the family of believers. Fear God. Honor the emperor. (1 Peter 2:11-17)

Notice that although this passage does echo the theme of Romans 13:1-5 in saying that the civil authorities are God's agents, "sent by him to punish those who do wrong and to praise those who do right," it focuses more on obeying civil authorities to maintain a status and reputation as honorable and law-abiding people among the Gentiles, so that they will thereby honor God and be drawn toward God. This draws on themes embodied in the quotes from the Gospels given and commented on above.

None of these passages states explicitly, "If you break the civil law, you are committing a sin against God"—though Romans 13:1-5 does come very close to saying exactly that.

However, these passages do establish a general principle and a commandment that the laws established by the civil authorities should be obeyed. This at least strongly implies that if we disobey civil authorities in matters that do not conflict with divine law, we are, in fact, disobeying divine law because we are disobeying the Biblical injunctions to honor and obey the civil authorities.

It is beyond the scope of the question to discuss exactly when it is legitimate to disobey civil authority. But the general answer is as suggested just above: where civil law clearly and egregiously conflicts with divine law, divine law must take precedence for the believer. By the same token, the believer must be ready and willing to take the consequences of disobeying civil authorities, which may mean social ostracism, fines, prison, or even death.