People can, and undoubtedly will, argue for some time about whether President Obama deserves the Nobel Peace Prize. Meanwhile, though, there's a simpler and more immediate question: Does the Constitution allow him to accept the award?
Article I, Section 9, of the Constitution, the emolument clause, clearly stipulates: "And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince or foreign State."
This sounds like a technicality, but if it is unconstitutional without explicit congressional consent, it is unconstitutional. We can't just decide that we will respect about some parts of the Constitution but not others because they aren't that important. Out of respect for the Constitution, Congress should pass a resolution authorizing Obama to accept the Prize.