Nobel

The Founders were (rightly) worried about the influence of foreign institutions on American policy. Clearly, there were concerns about foreign governments (see the great fear the Jeffersonians had that Hamilton and the Federalists were in league with Britain). But there were also concerns about other organizations as well. The primary example was the Catholic Church, with the long history of behind the scenes attacks against the British throne still fairly fresh in the collective memory.
Article I, Section 9 of the Constitution was drafted specifically to reduce the chances of foreign influence:

No title of nobility shall be granted by the United States: 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.

We have traditionally focused on the "present" in this sentence, as the titles of nobility were diminishing in value even as the Constitution was drafted. However, the concern of foreign influence is as real today as it was 220 years ago. The Nobel was awarded in an attempt to influence American foreign policy. Europe is attempting to move towards greater influence in world affairs, and key to that goal is a diminished America. By encouraging the weakness of Obama's foreign policy, they seek to strengthen themselves.

I do not suggest Obama is required to (or even should) decline the prize. While declining in a forceful manner could have improved his strength at home and abroad, he chose to accept. However, he must guard against allowing the award to influence his decisions in Afghanistan, Iraq, North Korea, and other troubled areas around the world. To allow the prize to enter into .his decision-making process would indeed be allowing a foreign body undue influence on American foreign policy