There is no basis to this claim. It is possible that some synthetic oils provide better mileage than some conventional oil, but the opposite claim is also true. I do believe that if you are in an extremely cold climate, synthetics might provide slightly better gas mileage because of better low temperature performance. However, once the engine warms up, the advantage disappears, so unless you are doing a lot of cold starts in a low temperature environment, I doubt you could see the difference.
The API test for qualifying for the energy conserving mark involves comparing the oil being evaluated to a reference synthetic oil. To get the energy conserving mark, an oil must provide better economy in the test engine (a Ford 4.6L I think) than the synthetic reference oil.
I personally have compared Mobil 1 5W30 to Havoline 5W30 in two vehicles. I ran both oils for many thousands of miles and then compared the cumulative fuel mileage (I keep a gas log) for each type. There was no difference (none, zero, zip, they were the same). I am in a relatively mild climate and do a lot of moderate to long trip driving, so someone that does a lot more short trip driving in relatively cold climate might have slightly different results. It is my opinion that you will not reduce fuel economy simply by switching from conventional to synthetic oil if you keep the same viscosity rating. Friction reducing additives are more likely to have an effect on fuel economy than just changing the base stock (assuming the viscosity is the same for the two base stocks).
Regards,
Ed White