What exactly does "President Obama will ‘fold faster than a lawn chair’" mean?

Solution 1:

A lawn chair is a lightweight piece of folding furniture, made out of aluminum or fiberglass. It is designed to be easy to fold up and put away, or to unfold and take out, when one has guests for a party on the lawn or patio.

In this case it means Obama will give in to pressure and abandon his position, if he has one, as he has often done in the past in the face of Republican pressure.

Solution 2:

Fold, meaning "give in to your opponents" probably comes from the game of poker, where you can literally fold your hand and put it down to signify giving up. A lawn chair is something that folds quickly, although in a different sense. So this is an idiom like

He lies like a rug.

which plays on the two different meanings of a word.