The constitution is imbued with the idea of rule of law, but I guess that you want a more "technical analysis," i.e., how the principle of "no one is above the law" can arise from the constitution from a less abstract and more palpable perspective.
Before I answer that question, I have to point out that, literally applying the law equally to everyone can achieve the same effect as the law grants privileges to some particular entities. A law that says "Mr. Thomas Paine shall be immuned from all punishment prescribed by federal laws" is an extreme example. But the line is not so bright. Laws can grant privileges to a class of entities and they can often arguably seen as putting them "above the law." In this sense, I'm not aware of constitutional principles that mandate such privilege granting is forbidden except the grant of nobility (Article I, Section 9, Clause 8: No Title of Nobility shall be granted by the United States). But such law exists, such as U.S. Code § 220506(a), which explicitly grants United States Olympic Committee some exclusive privileges, and is challenged in San Francisco Arts & Athletics, Inc. v. United States Olympic Committee, 483 U.S. 522 (1987).
Another relevant point is the President's pardon power. Article II, Section 2 reads: "The President ... shall have Power to grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment." States also grant similar powers to their respective governors. While the President and governors may not be above the law, can they put someone else above the law?
Putting these aside, let's assume that the law is "fair" and there is no power to pardon. Under such assumptions, a person is free of law, either because (a) the executive branch does not seek punishment or seek less heavy punishment after this person (other than civil suits), or (b) the judicial branch does not properly apply the law whenever this person is the defendant. Under these assumptions, I believe that the following two constitutional provisions provide the best support for mandating "no one is above the law", because such "free outlaw" won't exists if these two constitutional provisions are complied:
(a) Article II, Section 3: "The President ... shall take Care that the Laws be faithfully executed..."
(b) Article III, Section 2: "The judicial Power shall extend to all Cases, in Law and Equity, arising under this Constitution, the Laws of the United States, and Treaties made..."