Tire width (and tire pressure) are controversial because it is extremely difficult to find meaningful data and because there are so many variables involved. Variables include both rider preferences, such as speed, comfort, price, and durability, as well as physical factors, such as tire width, weight, types of rubber, casings, road surfaces, etc.
The issues that everyone agrees on are that narrower tires have less weight, less aerodynamic drag, and less comfort than wider tires. Weight includes both the tire itself (about 100 g per tire to switch from 23mm to 28mm) and the larger tube to go with it (about 25 g per tube). That gives a total of about 250 g, which for acceleration is equivalent to about 500 g on the frame. Drag is harder to quantify, but a thinner tire is more aerodynamic than a wider tire. Comfort is also hard to quantify, but a wider tire and lower pressure are generally more comfortable.
Rolling resistance is a contentious issue. Lennard Zinn wrote two columns about rolling resistance: Seriously, wider tires have lower rolling resistance than their narrower brethren and Again, bigger tires roll faster, citing data and comments from Schwalbe, Wheel Energy, and Zipp. Critics such as Lloyd Chambers (in Are Wider Road Bike Tires Faster) have noted that such comparisons assume that tires assume equal tire pressure for tires of different widths, ignore aerodynamics and handling, and assume unrealistically low loads (50 kg). It seems that wider tires probably have lower rolling resistance, but this position does not seem to have been thoroughly proven yet.
Whatever the theoretical ideal, there are a few critical issues that make wide tires less practical for many cyclists. First, most road bikes are designed for 23mm tires and may not have clearance for 27mm or wider tires. Second, most road tires sold are 23mm, and wider tires may be hard to find and more expensive. Third, most wide tires are designed for utility bike use and are heavier and less supple than racing tires. Fourth, wide tires may be less stable on rims designed for narrow tires (according to Zipp).
There are some indications that wider tires that are specifically designed for speed may have lower rolling resistance than thinner tires, but this is just one of many important factors that cyclists must consider.