Short answer:
Paul is certainly not disavowing the kosher laws! He is urging the Jewish believers to reject the strange doctrines and arguments being used to sow confusion.
He likens doctrines to meat, and the believers to priests. The Aaronic priests while in service at the Temple eat certain meats brought in sacrifice. Believers in Jesus / Yeshua are also priests of a separate order (Melchi-Tzedek), and therefore have meat to eat (the New Testament doctrines of salvation through repentance) that are different and higher than those of the Aaronic priests (the Temple sacrifices).
In no way is Paul saying the old laws are passed away, nor is he saying that the Aaronic priesthood has no use any more.
Long answer:
The passage from 1 Corinthians you quoted uses some of the same words but is not really related. I will focus on Hebrews here.
Hebrews 13 is the closing of the letter and Paul is basically making some final remarks. The outline of the chapter is almost of seemingly unrelated bullet points:
- Let brotherly love continue.
- Be not forgetful to entertain strangers...
- Remember them that are in bonds, as bound with them...
- Marriage is honourable in all...
- Let your conversation be without covetousness...
- Remember them which have the rule over you, who have spoken unto you the word of God...
- Be not carried about with divers and strange doctrines...
- Obey them that have the rule over you, and submit yourselves...
- Pray for us...
- Now the God of peace ... make you perfect in every good work to do his will...
- Suffer the word of exhortation...
- Know ye that our brother Timothy is set at liberty...
- Salute all them that have the rule over you...
- Grace be with you all. Amen.
The bolded point he spends more time on, verses 9 to 16. All this to demonstrate, verses 9 to 16 constitute the complete immediate context. However, by reference this section may tie itself to earlier sections of the letter. There is some correspondence to chapter 9, but not exactly I think.
Verse by verse, using King James Version 1769 text:
Be not carried about with divers and strange doctrines.
This sets the point of reference for what follows. We must remember at all times that he is talking about doctrines, the true ones versus the 'strange' ones being proffered to the Jewish believers.
Paul does not go into detail of what exactly he is responding to. But we can infer that this section serves as a refutation for these 'strange' doctrines based on the scriptures.
For it is a good thing that the heart be established with grace;
Grace is the means by which we may approach God in Spirit.
A key understanding here is that in Hebrew, korban is usually translated as "sacrifice" but it literally means "approach" or "come near". The only way to approach God is in sacrifice. (Exodus 23:15 "... and none shall appear before me empty[-handed].")
Hebrews 9 goes into an extensive discussion and comparison of the fact that the Aaronic priests can only approach the divine presence through animal sacrifice, but Jesus / Yeshua has sacrificed himself, and his blood is more pure than that of a bull or goat or red heifer. His sacrifice makes it possible for us to approach God without entering the Temple.
not with meats, which have not profited them that have been occupied therein.
This seems to be the critical confusion at issue. 'Meats' here is a reference to a sacrificial animal, not to the kosher dietary laws.
The sacrifices do not "establish the heart", their purpose is to atone for sins. Only God's grace is sufficient; otherwise the sacrifice of Jesus / Yeshua would have been unnecessary. Paul touches on this theme in several of his letters.
In many cases the Aaronic priests do eat the sacrificed meat, and offered grains, partly because the law prohibits them from owning their own land or cattle. But consumption of the meat is not the focus here; the sacrifice of the animal is the focus.
Animal sacrifice "has not profited them" for reasons discussed in the earlier chapters. Animal sacrifice only "covers" sin, it does not "remove" sin; and animal sacrifices must be repeatedly offered. This is not to say it is pointless! To claim the animal sacrifices are pointless is to mock the ordinances of God! But the sacrifice of Jesus / Yeshua is more potent because with that one sacrifice the sins of many are removed.
Also, as another answer points out, your International Standard Version contains here a deceptive distortion of scripture! Your translation says "food laws" but in the Greek it is only the plural "meats" (sometimes translated "foods"). The Greek does not say "laws" or "instructions" here. The translator has inserted his own opinion into the work, willfully distorting the Greek text! Get another Bible to compare with. I recommend the generally excellent New King James Version, which says "foods" here - accurate, but in my opinion not as clear as "meats".
We have an altar, whereof they have no right to eat which serve the tabernacle.
The believers in Jesus / Yeshua, including the Jewish ones, constitute a separate priestly order of Melchi-Tzedek, of which Jesus / Yeshua is now the High Priest forever (Psalm 110:4). Aaronic priests who are not believers in Yeshua are excluded from the Melchi-Tzedek priesthood and cannot approach God in the same way.
For the bodies of those beasts, whose blood is brought into the sanctuary by the high priest for sin, are burned without the camp.
A reference to Leviticus 4. The atonement for unintentional sin is to bring a bull, or goat, or ewe in sacrifice, and the bulk of the carcass must be burned to ashes outside the boundary of the camp - implying that in sacrificing it, the carcass becomes unclean as it is burned. (None of the meat is to be consumed by any priest.)
Wherefore Jesus also, that he might sanctify the people with his own blood, suffered without the gate.
Paul argues that the suffering and death of Jesus / Yeshua follows the type of the sin offering, and also serves as an atonement for sin.
Let us go forth therefore unto him without the camp, bearing his reproach.
Jewish believers in Jesus / Yeshua may approach God's presence without having to enter the Temple, bearing the blood of Yeshua instead of an animal sacrifice.
For here have we no continuing city, but we seek one to come.
Jerusalem is the earthly place within the Land of Israel where God chose to establish his divine presence in the Temple. In olam haba (the world/age to come), there will be a 'new Jerusalem' (Revelation 22). Side note, it is interesting that Paul references the city to come decades before John received the Revelation!
By him therefore let us offer the sacrifice of praise to God continually, that is, the fruit of our lips giving thanks to his name.
Finally we learn what we are to offer as we approach God's presence - our praise.
But to do good and to communicate forget not: for with such sacrifices God is well pleased.
Our good deeds and fruitful relationships are also suitable to present to God as we approach His presence.
And so we see, in detailed analysis, we find nowhere any indication that Paul seeks to discard the old laws. (And the dietary laws are actually not relevant, because the discussion is of sacrifice.) Rather, the laws are critical to understanding how we relate to God in the New Covenant!
