How to treat a hot oil burns on hands and leg?

How serious?: This can be very serious depending on whether it's 1st, 2nd or 3rd degree and what extent of area is burned. You absolutely should see your doctor unless it's trivial. If trivial, put antibiotic ointment on it, a telfa dressing, and analgesics for pain.

Immediately !: Remove yourself from the source of heat. Immerse your hands into a cool / room temperature water filled bucket. Pour cool / warm temp water onto your leg. Do not rub the burn areas. Call 911 and get transported immediately to the ed. You may develop second degree burns, and depending upon the amount of skin affected, you may have to go through long time of care by burn specialists.

It depends: On the severity of the lesion (degree) and surface affected (percentage of the skin). Combining both, it may range from little intervention (some topical preparations and/or gauze, for example) to immediate transfer to ICU or specialized centers, including eventually life-support measures. Each case should ideally be evaluated and triaged in an er, to determine what is necessary.