Exegesis of Deu. 13:1
The issue with Deu. 13:1-4 as cited in your original post is that the chapter doesn't actually begin with Deu. 13:1 (which is numbered as Deu. 12:32 in most Christian bibles). Rather, it begins with Deu. 12:29 in Christian bibles (and Jewish bibles).1
By starting with Deu. 12:32 (13:1), one neglects to appreciate the preceding passage which establishes the context of Deu. 12:32 (13:1).
In Deu. 12:29, it is written,
When Yahveh your god cuts off the nations from before you, [the nations] which you come there to inherit them, and you inherit them and dwell in their land,
כִּי יַכְרִית יהוה אֱלֹהֶיךָ אֶת הַגּוֹיִם אֲשֶׁר אַתָּה בָא שָׁמָּה לָרֶשֶׁת אוֹתָם מִפָּנֶיךָ וְיָרַשְׁתָּ אֹתָם וְיָשַׁבְתָּ בְּאַרְצָם
So, the Israelites have not yet entered the promised land, the land of Canaan. Yahveh is going to cut off (i.e., eliminate) the nations from those lands, and the Israelites will come and inherit those lands and dwell therein.
In Deu. 12:30, it is written,
Be careful lest you be lured after them, after their destruction from before you, and lest you ask about their gods, saying, "How did these nations serve their gods? Then I will do likewise!"
הִשָּׁמֶר לְךָ פֶּן תִּנָּקֵשׁ אַחֲרֵיהֶם אַחֲרֵי הִשָּׁמְדָם מִפָּנֶיךָ וּפֶן תִּדְרֹשׁ לֵאלֹהֵיהֶם לֵאמֹר אֵיכָה יַעַבְדוּ הַגּוֹיִם הָאֵלֶּה אֶת אֱלֹהֵיהֶם וְאֶעֱשֶׂה כֵּן גַּם אָנִי
Notice how Yahveh cut off the nations out of the lands that the Israelites were to inherit (v. 29). Why then would any Israelite ask how those very same nations that were cut off served their gods? And, not only that, but the Israelite says that he would serve Yahveh in the same manner that the nations served their gods: "so then I will do likewise!" (וְאֶעֱשֶׂה כֵּן גַּם אָנִי). Moses cautions the Israelites NOT to be lured after those nations by asking how they served their gods and then by serving Yahveh in the same manner.
In Deu. 12:31, it is written,
You shall not do likewise to Yavheh your god! For they did to their gods every abomination that Yahveh hated! For they have even burnt their sons and their daughters in fire to their gods!
לֹא תַעֲשֶׂה כֵן ליהוה אֱלֹהֶיךָ כִּי כָל תּוֹעֲבַת יהוה אֲשֶׁר שָׂנֵא עָשׂוּ לֵאלֹהֵיהֶם כִּי גַם אֶת בְּנֵיהֶם וְאֶת בְּנֹתֵיהֶם יִשְׂרְפוּ בָאֵשׁ לֵאלֹהֵיהֶם
Rather than permitting the Israelite who says, "Then I will do likewise!", Moses prohibits the Israelite, saying, "You shall not do likewise to Yahveh your god!" The reason, of course, is that the nations who inhabited the lands committed abominable acts to their gods, the very abominable acts which Yahveh hates, and for which He cut them out of the land (cp. Lev. 18:27; Deu. 18:12).
But again, why would the Israelites have even considered doing those acts for Yahveh? The reason is because the Israelites believed that Yavheh did not hate the acts themselves, but rather, they believed He hated the fact that the nations did those acts for their gods but not for Yahveh. So, by doing those acts for Yahveh, the one true god, the Israelites believed that Yavheh would approve of their actions. As Moses explains, those very acts themselves are abominations to Yahveh! With that context in mind, Deu. 12:32 makes more sense.
In Deu. 12:32, it is written,
All this word that I command you, observe to do it. Do not add to it nor diminish from it!
אֵת כָּל הַדָּבָר אֲשֶׁר אָנֹכִי מְצַוֶּה אֶתְכֶם אֹתוֹ תִשְׁמְרוּ לַעֲשׂוֹת לֹא תֹסֵף עָלָיו וְלֹא תִגְרַע מִמֶּנּוּ
The Israelites wanted to know how the nations served their gods so they could add those same practices to how they served Yahveh, believing that Yahveh would view them favorably as long as they did those practices for Yahveh. But, Moses says that those acts and practices were all abominations. Therefore, he warns them not to add to or dimish from Yahveh's commandments by infusing their practices with the practices of the nations that were cut off because of their abominations. Yahveh commands the Israelites how He expects them to serve Him; the Israelites are not to add to or dimish from those commandments by incorporating the practices of the nations that were cut off.
That being said, Deu. 12:32 (13:1) does not prohibit someone (i.e., another prophet) from adding other commandments for the Israelites to obey at a later time. Of course, such a person cannot change those commandments already commanded by adding to or diminishing from them.
Even Jews today have added to the Torah as a whole. For example, Exo. 12:2 (cp. Exo. 13:4) states that Aviv (אָבִיב) shall be the first month of the year. Yet, today, Jews use names derived from the Babylonian captivity, with the first month of the [agrictultural/festival] year being named Nisan (נִיסָן) instead of Aviv (cp. Est. 3:7).
Furthermore, the script that Torah scrolls are written in today (Ashuri script) is not the same script that Moshe would have written the Torah. Therefore, the Torah that we have today couldn't possibly be the same exact Torah Moshe was given on Mount Sinai.
Jewish Commentary on Deu. 13:1
Some Jewish rabbis used Deu. 13:1 as proof that the Torah was immutable and nothing could ever be added to it or diminished from it. One such rabbi was Moshe ben Maimon or Maimonides.2 Yet, there were other rabbis such as Yosef Albo who did not believe that Deu. 13:1 could be used as proof of the immutability/eternality of the Torah of Moshe.
In Sefer ha-Ikkarim, Book 3, Chapter 14, pp. 120-122, he wrote,
The opinion of Maimonides is that the Torah will never change in whole or in part. Hence one of his principal dogmas is the immutability of the Torah.
Maimonides bases this dogma upon the scriptural passage, "Thou shalt not add thereto, nor diminish from it."
The prohibition against adding to or subtracting from the commandments has no reference to the number of the commandments, but to the manner of performing them. The meaning of the prohibition is that we should not invent of our own mind or borrow from the idolaters some addition or dimunition in the manner of performing the commandments, because we may think it does greater honor to God. The context proves this is the meaning.
In section "Reeh" the Bible warns against idolatry: "When the Lord thy God shall cut off the nations from before thee..."
Now there is something strange in the sequence of thought in the above passage, "Take heed to thyself that thou be not ensnared to followe them, after that they are destroyed from before thee." Would a person be likely to be ensnared to follow them after he has seen them destroyed? Why then does the text connect the idea of inquiring after their gods with their destruction?
The Bible is concerned that one should not be enticed in any manner after idolatrous practices. Therefore it says, a person might make a mistake and suppose that the reason God commanded the destruction of the idolaters is because they offered to the idols the honorable service that is becoming to God. (Translation by Isaac Husik)
Jews keeping the Torah of Moses
Whether Jew or Gentile, no Christian is obligated to keep the Torah of Moses, for the Torah of Moses is the Old Covenant, and Christians, whether Jew or Gentile, are under the New Covenant.
Brothers, don't you know (for I speak to those who know the law) that the law is lord over a man as long as he lives? (Rom. 7:1)
Since Jews who believe in Christ have died with Christ (cp. Rom. 6:8), they are a new man and no longer under the law; the law is no longer their lord or master. For, the apostle Paul wrote,
so that, my brothers, you have also become dead to the law by the body of Christ, in order for you to become another man's, [even] he who was raised from the dead, so that we may bring forth fruit to God. (Rom. 7:4)
Even the rabbis wrote that a soon as a man dies, he is free from the commandments (i.e., of the Torah).3
1 Jewish bibles stop chapter 12 of Deuteronomy after thirty-one (31) verses.
2 Introduction to the Mishna, Tractate Sanhedrin, Chapter Ten, Ninth Principle of the Faith; Mishneh Torah, Sefer Madda, Hilkhot Yesodei ha-Torah, Chapter Nine, Halakha 1
3 Babylonian Talmud, Seder Tohorot, Tractate Nidda, Chapter 9, Folio 61b: כיון שמת אדם נעשה חפשי מן המצות
Sefer ha-Ikkarim (The Book of Principles). Albo, Joseph. Trans. Husik, Isaac. Philadephia: Jewish Publication Society, 1946.