Maimonides writes:²

The Rabbinical Courts maintain the right to issue decrees and forbid that which is [biblically] permitted, and these prohibitions stand for perpetuity. They are also entitled to temporarily lift Torah prohibitions.³ So what is the meaning of the Torah's prohibition: "You shall neither add to it, nor subtract from it"?

[Rather, the intent of this prohibition is that we] not add on the words of the Torah nor subtract from them, and permanently establish [the addition or subtraction] as part of the Scriptures. This [prohibition] applies both to the Written Law as well as the Oral Tradition [transmitted to Moses on Mount Sinai].

For example: It is written in the Torah,⁴ "Do not cook a kid in its mother's milk." The Oral Tradition explains that this verse forbids cooking and consuming meat and milk -- both the meat of domesticated and undomesticated animals -- but the meat of fowl may be eaten with milk according to the law of the Torah.

If a Beth Din will arise and will permit the consumption of undomesticated animals with milk -- this is subtracting [from the words of the Torah]. If the beth din forbids the consumption of the meat of fowl [together with milk], saying that it is included in the word "kid" -- this is adding [on the words of the Torah].

If, however, the [Beth din] says, "the meat of fowl is permitted [together with milk] according to the Torah, however we are forbidding it, and will notify the public that this is a decree which will prevent [the following catastrophe]: People might say that '[just as] meat of fowl is permitted because it is not stated explicitly in the verse, so too the undomesticated beast is permitted because it, too, is not stated explicitly.' And another might say, 'also the meat of a goat is permitted with the milk of a cow or sheep, for only a mother of the same species is mentioned.' And yet another will say, 'also the meat of a goat with goat milk which is not from its own mother is permitted, for the Torah speaks of [cooking meat in] its own mother's [milk].' Therefore we are prohibiting all meat with milk -- even meat of fowl." Such a decree is not adding on the Torah, rather it is creating a fence surrounding [and protecting] the words of the Torah.

To summarize:

The prohibition only applies only if a person were to come along and say that G-d told him to add another commandment to, or subtract one from, the existing 613.