Yes, all food is permitted for Christians
Yes, according to the New Testament it is perfectly okay for Christians to eat pork and shellfish, both of which were prohibited to Jews in the Law of Moses. This is illustrated by Mark’s comment that Jesus “declared all foods clean”:
Mark 7:18 And [Jesus] said to them, “Then are you also without understanding? Do you not see that whatever goes into a person from outside cannot defile him, 19 since it enters not his heart but his stomach, and is expelled?” Thus he declared all foods clean.
Not only does the New Testament allow all food, Paul specifically warns against those who try and restrict Christians from certain foods:
1 Timothy 1 : 1 Now the Spirit expressly says that in later times some will depart from the faith by devoting themselves to deceitful spirits and teachings of demons, 2 through the insincerity of liars whose consciences are seared, 3 who forbid marriage and require abstinence from foods that God created to be received with thanksgiving by those who believe and know the truth. 4 For everything created by God is good, and nothing is to be rejected if it is received with thanksgiving, 5 for it is made holy by the word of God and prayer.
It might sound extreme for Paul to give such a strong warning against those who promote food restrictions, but in the context of the 1st Century the food restrictions were often part of a package of imposing other things from the Law of Moses on Gentiles such as Sabbaths and even circumcision.
Provided eating is not done in front of those whom it might damage
The only exception to the general freedom above is where any food — either a “clean” food offered to idols (e.g., beef or mutton), or a food that was unclean to Jews like pork or shellfish, or by extension today beef to a Hindu or meat to a Buddhist — will cause a stumbling block or barrier between someone and belief in Christ.
Paul discusses how and why not to offend converts from idol-worship in 1 Cor. 10, and how not to offend converts from Judaism in Rom. 14. However, in those chapters Paul repeatedly makes it clear that, apart from not causing other believers (particularly new believers) to stumble, he himself is free to eat anything, and so are Christians.
Paul specifically warns about Christians who try to establish again the Jewish food laws:
1 Timothy 4:1 Now the Spirit expressly says that in later times some will depart from the faith by devoting themselves to deceitful spirits and teachings of demons, 2 through the insincerity of liars whose consciences are seared, 3 who forbid marriage and require abstinence from foods that God created to be received with thanksgiving by those who believe and know the truth. 4 For everything created by God is good, and nothing is to be rejected if it is received with thanksgiving, 5 for it is made holy by the word of God and prayer.
The rules against pork and shellfish in the Law of Moses were a temporary restriction. Now circumcision, and the Law of Moses generally, has been fulfilled and finished. The following two passages argue from the abolition of circumcision to prove abolition of all the Law.
Ephesians 2:11 Therefore remember that at one time you Gentiles in the flesh, called “the uncircumcision” by what is called the circumcision, which is made in the flesh by hands— 12 remember that you were at that time separated from Christ, alienated from the commonwealth of Israel and strangers to the covenants of promise, having no hope and without God in the world. 13 But now in Christ Jesus you who once were far off have been brought near by the blood of Christ. 14 For he himself is our peace, who has made us both one and has broken down in his flesh the dividing wall of hostility 15 by abolishing the law of commandments expressed in ordinances, that he might create in himself one new man in place of the two, so making peace, 16 and might reconcile us both to God in one body through the cross, thereby killing the hostility.
Colossians 2:13 And you, who were dead in your trespasses and the uncircumcision of your flesh, God made alive together with him, having forgiven us all our trespasses, 14 by canceling the record of debt that stood against us with its legal demands. This he set aside, nailing it to the cross. 15 He disarmed the rulers and authorities and put them to open shame, by triumphing over them in him. 16 Therefore let no one pass judgment on you in questions of food and drink, or with regard to a festival or a new moon or a Sabbath.
Note above that Paul also dismisses the Sabbath itself along with rules about food and drink.
Appendix : Food with blood
This has been discussed previously:
The idea of clean and unclean animals dates back to Noah selecting two of each kind of the unclean animals and seven of each kind of the clean animals for the ark. After the ark came to rest, God removed any restriction on eating meat:
Genesis 9:1 And God blessed Noah and his sons and said to them, “Be fruitful and multiply and fill the earth. 2 The fear of you and the dread of you shall be upon every beast of the earth and upon every bird of the heavens, upon everything that creeps on the ground and all the fish of the sea. Into your hand they are delivered. 3 Every moving thing that lives shall be food for you. And as I gave you the green plants, I give you everything. 4 But you shall not eat flesh with its life, that is, its blood. 5 And for your lifeblood I will require a reckoning: from every beast I will require it and from man. From his fellow man I will require a reckoning for the life of man.
Thus leaving only the warning about meat with the blood in it. Blood is a separate issue that was binding on Noah before the Law of Moses, and also more or less binding on Christians in the Jerusalem and Antioch areas during the early years of the church. In Acts 15, as a result of a dispute with Jewish Christians who had very extreme views in Antioch demanding circumcision, wanting Gentile Christians to adopt all the food laws, etc., James and the Jerusalem Church adopted a compromise position with only three food requirements:
Acts 21:25 But as for the Gentiles who have believed, we have sent a letter with our judgment that they should abstain from what has been sacrificed to idols, and from blood, and from what has been strangled, and from sexual immorality.”
However Paul later overturned these three rules about meat offered to idols, meat with blood in it, and strangled meat as well:
1Co.10:25 Eat whatever is sold in the meat market without raising any question on the ground of conscience. 26 For “the earth is the Lord’s, and the fullness thereof.”
This rule applied to beef and mutton too, not just pork.