There are some interesting references to paper, origami, cards, etc. in House of Cards. Perhaps for Claire, it's a reference to two things - one being her suppressed creativity and how that ties in to her longing for Adam. We don't know that much about her past, except that which has been revealed through her conversations with Adam. We also learn a little about her choices when she sat by Steve's bedside as she discusses the conscious decision she made to be with Frank, not necessarily for love, but for the respect and freedom he would give her. Ironically, though Claire is not locked into a life of being the trophy wife, she is seemingly trapped in a marriage of partnership and reason. Adam (and the origami) represent freedom, art, and impermanence. All that which she does not possess.
The second reference may be to the constant folding and unfolding of events in her life. As close as she may be to Frank, he is still able to use her as a pawn to further his own goals. She makes decisive efforts to curtail this when she sabotages the water bill.
Paper is temporary; it does not provide real shelter, as was mentioned by one of the workers at the shipyard meeting Peter Russo held. Nor does it protect us from the elements. It can all collapse in a moment, just as a structure made of cards will do. Money, made of paper, can be easily burned, destroyed or lost. His kids also do origami - a foreshadowing of the two tiny victims who will pay a huge price for Frank's lust for power.
The homeless guy took a bill - a physical representation of what our society deems to be worth something, folded it, and turned it into a piece of art for Claire. It wasn't a small amount for a homeless person - it was $20! A blatant reminder that those things which are valuable to some, may mean nothing to others. It was an eerie reminder that her life and everything she values can quickly fall apart.
Lastly, Frank in his speech when he was honored at his alma mater talks about harmony, and how "it is not about what's lasting or permanent, it's about individual voices coming together for a moment." This is so contradictory to everything Frank truly stands for. The reality is, he backstabs, manipulates, and deceives everyone around him, especially those who are near him. The only one he does not deceive is Stamper (which should be discussed on a separate thread as that is an interesting theme as well) Frank is fully driven by the hunger to establish legacy, permanence, and power - basically the opposite of what origami represents.
I would love to hear more feedback from others - I think this is a really interesting concept and there is so much room to dig deeper.