Yes, we should obey our authorities and show respect to them. Let everyone 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 judgement on themselves. 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 doesn't bear the sword in vain; for he is God's minister, an avenger to execute God's wrath - on him who practises evil. Therefore you must be subject to authority, not only because of wrath, but also for conscience sake. For because of this, you also pay taxes. Render therefore to all their due taxes to whom taxes are due, customs to whom customs, fear to whom fear, honour to whom honour (Rom. 13:1-7). Almost the same teaching is given by Peter (Cfr. 1 Pet. 2: 13-14, 19). Jesus had already taught "Render therefore to Caesar the things that are Caesar's, and to God the things that are God's" (Mt. 22:21) Cfr. Ecc. 8:2.
However, when the ruler commands us to do something contrary to our faith, we are not bound to do. Because God is the supreme ruler who is not to be disobeyed for the sake of mundane rulers. When the Sanhedrin commanded Peter and John not to preach in the name of Jesus, their reply was that they had to listen to God more than to them (Acts 4:18-19). We ought to obey God rather than men" (Acts 5:29).