Is driving over the speed limit for any reason sinful? What if someone says that they can drive over the speed limit if it helps them to get to worship service on time. Or what if there's an emergency and someone needs to speed to the hospital? I see that Romans 13 tells us to be subject to the governing authorities and our states in America have posted speed limits, so I assume we are to obey the laws of the land.
"Let every soul be subject to the governing authorities. For there is no authority except from God, and the authorities that exist are appointed by God. Therefore whoever resists the authority resists the ordinance of God, and those who resist will bring judgment on themselves" (Romans 13:1-2).
Throughout the New Testament, there is a concept that there is a chain or hierarchy of authority. The ultimate authority is God and He delegates a portion of His authority to others to carry out His will. In this case, the government exists to punish evildoers. "For rulers are not a terror to good works, but to evil. Do you want to be unafraid of the authority? Do what is good, and you will have praise from the same. For he is God's minister to you for good. But if you do evil, be afraid; for he does not bear the sword in vain; for he is God's minister, an avenger to execute wrath on him who practices evil" (Romans 13:3-4).
So long as the government is in subjection to God, we are required to be in subjection to the government.
"Therefore submit yourselves to every ordinance of man for the Lord's sake, whether to the king as supreme, or to governors, as to those who are sent by him for the punishment of evildoers and for the praise of those who do good. For this is the will of God, that by doing good you may put to silence the ignorance of foolish men -- as free, yet not using liberty as a cloak for vice, but as bondservants of God. Honor all people. Love the brotherhood. Fear God. Honor the king" (I Peter 2:13-17).
Notice that our obedience to the laws is a part of our influence on the world around us. For example, if I generally am a law-abiding citizen and well respected in my community, then when I do take a stand to say, for instance, that homosexuality is wrong and the government should not support it, my statement has a bigger impact than a person who is always rebelling against the government and break rules right and left.
Now if the government issued a rule that violates God's laws, Christians recognize that the government only exists by God's authority. We have a responsibility to follow the higher authority. When the Jewish court issued a ruling that the apostles were to stop teaching in the name of Jesus, "But Peter and the other apostles answered and said: 'We ought to obey God rather than men'" (Acts 5:29).
Another approach to this issue is addressed in "The Reason for Regulations."
In your examples, you are making a common mistake. Many people think that if exceptions exist to a rule, then the rule isn't necessary. This is the way a lot of Baptists argue against baptism. Some are honest enough to admit that God does command baptism, but because they can imagine exceptions that they think God will find acceptable, such as a person dying before they can get to the water to be baptized, they feel the law can be ignored all the time. So let's look at the excuses for speeding.
I need to speed in order to get to services in time
Do the math sometime. Many years ago a fellow worker told me she had to go 15 miles-per-hour over the speed limit in order to get to work in time. I asked her how far she drove and then did the math and showed her that her high-speed antics (with a child in the back seat!) saved her all of 45 seconds off her trip! Say it is a 6 minute trip to the building at 45 mph. If you went 60 mph it would become a 4.6 minute trip. And that's not taking into account traffic lights, stop signs, and varying speed. You saved all of 1.4 minutes by becoming a traffic hazard! Are 1.4 minutes going to make that much of a difference?
But the reality is that the whole situation is self-made. If you left the house 2 minutes earlier you would arrive at about the same time without speeding. In other words, you are justifying breaking a law because you were careless with your time.
Is this any different than what the Jews did? They played one law off another to excuse themselves from following laws they didn't like. "He answered and said to them, "Why do you also transgress the commandment of God because of your tradition? For God commanded, saying, 'Honor your father and your mother'; and, 'He who curses father or mother, let him be put to death.' But you say, 'Whoever says to his father or mother, "Whatever profit you might have received from me is a gift to God" -- then he need not honor his father or mother.' Thus you have made the commandment of God of no effect by your tradition" (Matthew 15:3-6). Do you see the similarity? Because "good" is being done, then this excuses breaking another law and Jesus soundly condemns this attitude.
There is an emergency and I need to get someone to the hospital
Even emergency vehicles technically don't speed. What they are allowed to do by the laws is to gain the right-of-way. Everyone is to move out of their way so they can get to their destination more quickly. This is actually more important than raw speed. Speed limits are still observed because the limits are for safety. Going fast doesn't help if you have an accident on the way.
But emergencies are often overlooked because "For judgment is without mercy to the one who has shown no mercy. Mercy triumphs over judgment" (James 2:13). In other words, breaking the law is still wrong, but the consequences are softened because we place mercy as a more important matter.
Rather than counting on mercy, we should take a moment to consider alternatives -- calling for an ambulance, calling the police, if we are far away from help offering to meet either of these part of the way. But remember always that the aim is to get help for someone fast. If you crash on the way you failed to do what you set out to accomplish.
Of course, these types of emergencies with no other alternatives are extremely rare. The fact that we might find a time when a police officer or a judge might overlook a law being broken because we were trying to save someone's life doesn't imply that in general the speed limit law can be ignored.
The problem is who is deciding that a law can be ignored? The one who decides that breaking the law benefits him. Is his judgment fair? Is it unbiased? Is it based on all the facts? Remember what Jeremiah said, "O LORD, I know the way of man is not in himself; it is not in man who walks to direct his own steps" (Jeremiah 10:23).