Is your oil leak new or has it been leaking for a while? If your crankcase breather is plugged, it could force an oil leak and oil burning at the same time. Does the oil (inside the crankcase) smell like fuel?
I second the idea of using ONLY the recommended lubricant and keeping the "genie in a can" inside the can it came in and on the store shelf. (do not overfill the oil) A stuck float or choke in the carburetor could add excess fuel to your oil causing it to leak/burn prematurely.
Think positive. Any oil burning that came, apparently, out of nowhere, can stop just as quickly. (once you find/repair the problem) If the genny starts and runs well and generates rated power, (and doesn't make unusual noises) it's probably not an internal mechanical issue. Check the air filter for oil saturation.
Chum lee