Can the President enter into a treaty that does away with the Constitution?
Article VI, Clause 2 of the Constitution declares that "This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land." At the time of the writing of the Constitution the Founders wanted to be sure that no state would have jurisdictional conflicts between the states and the federal government, at least limiting it. So they declared that the Constitution is the supreme law of the land. All federal laws that were in accordance with the Constitution were also superior to state laws.
The Founders also wanted to be sure that state laws would not conflict with the treaties that the nation entered into. Such conflicts or potential conflicts the Founders realized would greatly weaken the nation's ability to negotiate treaties. Just as Clause 2 does not imply that federal laws are superior to the Constitution it does not imply that treaties are superior. The Supreme Court declaed that a treaty cannot obligate the violation of Constitutional grarantees, such as those found in the Bill of Rights. (see Reid v. Covert 1957)
Furthermore, since a treaty is to be considered on the same level as any other federal law then Congress can repeal the treaty as they can repeal a law.
Under Supreme Court precedents, the last expression of the sovereign will controls, so an act of Congress that is in conflict with a treaty will control if the act became law after the Senate ratified the treaty, and vice versa. (Gary Lawson, The Heritage Guide to the Constitution  , pg. 294)