What is the biblical basis for dual-covenant theology? Dual-covenant theology, or two-covenant theology, holds that there are two covenants in effect—one, the Old Covenant (with the Mosaic Law) applying to Jews, and the second, the New Covenant (through Jesus), applying only to Gentiles.

I've heard some dismiss this doctrine as a politically correct attempt to make Christianity "less antisemitic."  But among its proponents, what are the biblical arguments made on its behalf?  

I'd like to see arguments from both Old and New Testaments, if possible.  If there are arguments that rely on translating particular verses differently than what is traditional, those would be great to include as well.
Since your question focuses on dual covenant theology's biblical basis, I will cite the main points and verses and leave you to research the major proponents' more complex arguments in their writings at your leisure:


After the Great Flood, God imposed a set of laws on Noah and his family and their descendants (Genesis 9:3-10), which makes them binding on all of humanity alive today (known as the "Noahide Code" or the "Noahide Laws"). These are the baseline requirements of behavior for all people.
God made a covenant with Abraham and his descendants through Isaac, i.e. the Israelite/Jewish nation (Genesis 17:4-14, 19). This covenant was promised by God to last forever (v.7). 
One major legal component of this covenant was that Abraham and his male progeny be circumcised (vv.10-14). God said that Abraham had obeyed "My commandments, My statutes and My laws" (26:5) under this covenant.
The Abrahamic Covenant was refined in greater detail at Mt. Sinai. A major component of the Covenant was the Mosaic Law (Exodus 19-24). The people accepted to obey the whole Law by saying "we will hear and we will obey" (24:7). Later, under Ezra, they communally reaffirmed their faithfulness to the Law (Nehemiah 9-10).
God promised that the Mosaic Law would remain in force forever (Exodus 27:21, 28:43, 29:28, 30:21, 31:17; Leviticus 6:18, 22; 7:34, 36; 10:9, 15; 17:7; 23:14, 21, 41; 24:3; Numbers 10:8; 15:15; 18:8, 11, 19, 23; 19:10; Deuteronomy 5:29; Psalm 119:160, etc), and is "perfect" (Psalm 19:7).
God said that any "prophet" who tells Israelites/Jews to stop obeying the Mosaic Law is a false prophet (Deuteronomy 13:1-6, 18).
God said that the elements of the Mosaic Law will not be changed even a little (Deuteronomy 4:2, 12:32, 13:1). This was supported by Jesus, who said that "neither a jot nor a tittle" (i.e. the letter yud or the calligraphic elaboration of a yud) will pass from the Mosaic Law until "all is fulfilled" (Matthew 5:17-19; Luke 16:17). This fulfilment he speaks of has not yet occurred, since the messianic age has not yet arrived, and we are still living in an unredeemed world.
God and the prophets said that the the Israelites/Jews were capable of completely obeying the Mosaic Law properly (Deuteronomy 30:9-14; Psalm 119; Luke 10:25-38; Matthew 19:16-20).
God said that the "New Covenant" will involve "My Law" (torati) being placed inside people and written on their hearts, implying that they will fully internalize the Mosaic Law and obey it fully and properly (Jeremiah 31:33-34). 
God promised that as long as the celestial bodies remain in place, Israel will remain a distinct nation (Jeremiah 31:35-37), not be absorbed and undifferentiated into a larger non-Jewish religious movement.
Jesus was only focused on delivering his message to "the house of Israel", not to Gentiles (Matthew 10:6; 15:24), even referring to them as "dogs" on one occasion (15:26-27). Jesus never told his (Jewish) followers that they no longer had to obey the Mosaic Law, he gave the Law a different interpretation from the scribes and Pharisees (e.g., grinding wheat in one's hands is not a Sabbath day violation, 12:1-5; miraculously healing the sick is not a violation of the Sabbath day, 12:9-13; eating with unwashed hands is not a violation of the kosher laws, 15:2, 20=Mark 7:2, 5, 18, etc).
"Nobody comes to the Father" except through Jesus (John 14:6), but Jews don't need to "come" to God because they're already with him, and have been since Abraham's time (e.g., Genesis 28:15; Exodus 33:14; Joshua 1:9; Jeremiah 29:13; Psalm 41:12; Zephaniah 3:17).
When Jesus gave the Great Commission in Matthew 28, he told his followers to spread the Gospel to "all nations". "The nations" (in Hebrew ha-goyim) is a common idiom referring to the non-Jewish world. E.g., Isaiah 9:1 which speaks of "Galilee of the nations" (Galil ha-goyim), is translated in the LXX and Gospels as "Galilee of the Gentiles" (Matthew 4:15). So Matthew 28:19 can be translated as 'go therefore and make disciples of the entire Gentile world, baptizing them ...' (=Luke 24:47, 'should be proclaimed in his name to the whole Gentile world, beginning from Jerusalem...'), not the Jewish people.
At the Jerusalem Council, James ruled that non-Jewish members of the Christian community would not be required to convert to Judaism and become bound by the Mosaic Law ("judaizing"). Instead, they were only required to follow the Noachide Laws that were already binding on all humanity (Acts 15:5, 19-21, 21:25). (Note that the exact number and content of the Noahide Laws were a matter of debate within the Jewish community for many centuries, with different groups having different lists and interpretations. So it's not unusual that James and the Council would have their own unique version.)
The Jerusalem Council, which was the authoritative leadership of the Jewish-Christian communities (Acts 15, 21:18; Galatians 2:9, 12), never authorized the reverse, "gentilizing" i.e. that Jews/Jewish-Christians can or must abandon any aspect of the Mosaic Law (Acts 21:20-21). (Levine refers to this as the "two-track method" of James, see book below).
Peter's vision where a voice tells him to "kill and eat" "unclean creatures" (Acts 10:10-16) is not related to kosher food laws - Peter himself interprets his vision as meaning "I should call no man unholy or unclean" (v.28), not food.
Paul was the "apostle to the Gentiles" (Galatians 1:15-16, 2:8; Romans 11:13; 1 Timothy 2:7), and so the authoritative leader of the Gentile-Christian communities. His letters were written to Gentiles only: e.g., the Galatians, the Corinthians, the Ephesians, the Romans, etc, not to Jews, and so have no bearing on what Jews/Jewish-Christians must to do/believe. One example of this is his detailed discussions about whether believers should eat meat from the marketplace (i.e. had most likely been sacrificed to idols before being sold in the agora) (1 Corinthians 8, 10); this is only relevant to Gentiles, since the Jewish community in every city had their own meat slaughtering facilities and markets in line with the Mosaic Law, and so this was never a problem for them.
While Paul strongly objected many times to Gentile-Christians "judaizing" (most prominently, by becoming circumcised) (e.g., Galatians 3), he does not state that Jews/Jewish-Christians must "gentilize". In fact, he acted in the reverse, circumcizing Timothy (whose mother was Jewish, making him a Jewish-Christian) (Acts 16:3) but not Titus, a Greek (Galatians 2:3), and paying for and joining a group of (Jewish) nazirites performing vows at the Temple (Acts 21:23-26), as the Jerusalem Council enjoined (v.23).
When Paul wrote, "there is no Jew or Greek, no slave or free, no male or female, for all are one in Christ Jesus" (Galatians 3:28), he was referring to the equality of all believers who are mystically one body, but this oneness does not erase the distinct social and religious roles of different people: e.g., women must still submit to men (1 Timothy 2:12; 1 Corinthians 14:34) and cover their heads (11:6), slaves must still submit to their masters (even if both are Christians) (Colossians 3:22; Ephesians 6:5; Philemon). Ergo, Jews are still required to obey the Mosaic Law while Gentiles are still only required to obey the Noahide Laws.
In Revelation, John of Patmos describes the great dragon making war on the rest of the "woman's" offspring, those who "keep the commandments of God and hold to the testimony of Jesus" (12:17). In Hebrew, the phrase "commandments of God" (mitzvot Elohim) invariably refers to the entirety of the Mosaic Law. In addition, the "mark" of the Beasts (13:16) appears to be a demonic parody of the wearing of phylacteries (tefillin) worn by Jews on the head and hand in obedience to Deuteronomy 6:8 and 11:18. This implies that the Jewish-Christians were still keeping (and John expected them to continue keeping) the Mosaic Law.
Paul cites the example of Abraham, who "believed God and it was credited to him as righteousness" (Genesis 15:6) as a Pentateuchal basis for salvation by faith/grace (Romans 4:3). But there is a second major method of "righteousness" (tzedekah) mentioned in the Pentateuch, when Moses tells the children of Israel that "this shall be our righteousness: to be careful to do all this commanded before the LORD our God, as he has commanded us" (Deuteronomy 6:25), i.e. obeying the Mosaic Law. Abraham availed himself of this second method in addition to the first (Genesis 26:5).


For more information, I recommend you read "The Rapprochement Between Jews and Christians" by Reinhold Niebuhr, "The Star of Redemption" by Franz Rosenzweig, "Jesus the Pharisee" by Harvey Falk, and "The Misunderstood Jew" by Amy-Jill Levine. Pope John Paul II stated agreement with some of the above tenets. Some other writers you may find useful to check out are: David Stern, Stan Telchin, Maimonides, Jacob Emden, Samson Raphael Hirsch, Moses Mendelssohn, Sir James Parkes, James Trimm, William Campbell, and Mark Nanos. Happy researching!