Do tube protectors increase rolling resistance or just add weight? I have been experiencing a lot of flats lately (proportional to the increase in time/distance I have been doing), so I am thinking of adding a tube protector (e.g., one of these) to each tire. 

Will this alter rolling resistance or just add weight to the wheel? I'm not sure what goes into rolling resistance aside from tire material and pressure. I expect a slight slow-down, but I'm not sure what to expect. I'm willing to trade slower speed for the time fixing flats.
It likely depends a lot on your tire pressure.  For high-pressure tires the liner just adds to the stiffness of the tire and might actually reduce rolling resistance slightly.

For lower pressure tires, the liner must flex and, to a degree, slide against the tire and tube with every revolution, increasing internally dissipated energy and hence rolling resistance.  But the effect would be slight.

A better solution is to use Kevlar (or equivalent) belted tires.  These have about 10x the puncture resistance of standard tires, have negligible effect on rolling resistance, and are much easier to deal with than the separate liner.  But of course they're 30-50% more expensive, and hard to find in certain sizes.

(NB: Don't confuse "Kevlar belted" tires with "Kevlar bead" tires -- Kevlar bead is used in folding tires, some of which are Kevlar belted and some not.  And I personally find folding tires a PITA to install the first time -- though some people love them.)