This question is extremely contentious because various people want to support a pre-existing theological position. Let us observe the following facts:
1. The NT quotes the OT laws (from the Torah) frequently as authoritative
Eph 6:2, 3 quotes Deut 5:16, Ex 20:12; James 2:11 quotes Ex 20:13; Rom 13:9 quotes Ex 20:13-15, 17; Rom 7:7 quotes Ex 20:17; Acts 23:5 quotes Ex 22:28; Heb 9:20 quotes Ex 24:8; 1 Peter 1:16 quotes Lev 19:2; Matt 22:39, James 2:8, Gal 5:14 quotes Lev 19:18; 2 Tim 2:19 quotes Num 16:5; Matt 19:18, 19 quotes Deut 5:16-20; Mark 12:32 quotes Deut 6:4; 2 Cor 13:1 quotes Deut 19:15; Matt 5:31, 19:7 quotes Deut 24:1; 1 Cor 9:9 Deut 25:4; Rom 12:19 quotes Deut 32:35; Heb 10:30 quotes Deut 32:35, 36, etc, etc.
2. Heb 8:13 does NOT say that the Law was abolished but says the Covenant was abolished.
Heb 8:13 - By speaking of a new covenant He has made the first one
obsolete; and what is obsolete and aging will soon disappear.
Now, notice why a new covenant was required:
Heb 8:8 - 12 - But God found fault with the people and said: " ... I will put My laws in their minds and inscribe them on their hearts. And I will be their God, and they will be My people. No longer will each one teach his neighbor or his brother, saying, ‘Know the Lord,’ because they will all know Me, from the least of them to the greatest. For I will forgive their iniquities and will remember their sins no more." [Quoting Jer 31:31-34]
Note further that the covenant was cancelled because the people (the Jews) were not faithful. The NT's "new covenant" involves writing laws on the hearts of the people. This is consistent with Matt 5:17-19 -
Do not think that I have come to abolish the Law or the Prophets. I
have not come to abolish them, but to fulfill them. For I tell you
truly, until heaven and earth pass away, not a single jot, not a
stroke of a pen, will disappear from the Law until everything is
accomplished. So then, whoever breaks one of the least of these
commandments and teaches others to do likewise will be called least in
the kingdom of heaven; but whoever practices and teaches them will be
called great in the kingdom of heaven.
Jesus then goes on the amplify the laws (mostly but not exclusively the 10 commandments and other moral requirements.)
3. The Law is praised in the NT frequently
“But it is easier for heaven and earth to pass away than for one jot of the Law to become void.” (Luke 16:17), the law is essential because “through the law we become conscious of sin” (Rom 3:21, 7:7, 13), “we uphold the law by faith” (Rom 3:31), “the law is holy, and the commandment is holy, righteous and good” (Rom 7:12), “the law is spiritual” (Rom 7:14), “the law is good” (1 Tim 1:8), keeping the law is to do right (James 2:8). “Do we, then, nullify the law by this faith? Certainly not! Instead, we uphold the law.” (Rom 3:31). “What then? Shall we sin because we are not under law, but under grace? Certainly not!” (Rom 6:15); “we are now slaves of righteousness” (Rom 6:16), or, “slaves to God” (Rom 6:22).
4. The NT contains many more "laws"
The NT contains more than 1000 laws that the Christian is bound by - more than the Torah. Some wit has even enumerated them in places such as https://www.abc.net.au/reslib/201407/r1308729_17984331.pdf and https://www.cai.org/bible-studies/1050-new-testament-commands and https://www.facebook.com/notes/the-real-truth-movement/the-1050-new-testament-commandmentslaws/690826194404882/ and http://www.wholebible.com/NT_commandments.htm and http://www.biblicalresearchreports.com/gods-commands-in-the-new-testament/
A careful examination of the NT laws shows that they are all based on the moral laws of the Torah.
5. The Levitical Covenant cannot be kept even if we wanted to
The Levitical Covenant (Lev 1-9, 16, 21-27 , Num 3, 4, 8, 18, 25:10-13, Deut 33:8-11, Neh 13:29, Mal 2:4-8) with its sacrifices, formal priesthood, Temple services, rites and ceremonies cannot be kept even if we wanted to - there is no priesthood, no temple and no sacrifices to offer. However, that was never part of the covenant with Israel (Ex 19-24) and was never part of the new covenant with Christians (Matt 26:28, Mark 14:24, Luke 22:20, 1 Cor 11:25, 2 Cor 3:6, Heb 8:6-13, 9:15, 10:16, 29, 12:24, Jer 31:31, 33). Jesus came to fulfill the Levitical covenant as our high priest in the heavenly temple (Heb 4:14-16, 7:23-28, 8:1, 2, 9:1-28, 10:1-18).
6. New Covenant
One of the best summaries of the New Covenant is found in 1 Peter 1, 2. Notice how often it alludes to the Torah and its moral laws:
- Purpose: “But you are a chosen people, a royal priesthood, a holy nation, God’s special possession, that you may express the praises of him who called you out of darkness into his wonderful light … Live such good lives among the pagans that, though they accuse you of doing wrong, they may see your good deeds and glorify God on the day he visits us.” 1 Peter 2:9, 12. (See also Matt 5:16.)
- The promise: Salvation by grace through the promised Messiah, 1 Peter 1:3-12, 20, and freedom from slavery to sin, 1 Peter 2:16. (See also 2 Peter 2:19.)
- Moral Requirements: holiness (1 Peter 1:15), Purity (v22), Obey the truth (v22), love (v22), “rid yourselves of all malice and all deceit, hypocrisy, envy, and slander of every kind” (1 Peter 2:1), abstain from sinful desires (1 Peter 2:11), submit to civil authorities (v13-17), see also Rom 13:1-7, etc.
- Sacrifice: Blood of Jesus, 1 Peter 1:18, 19