To burn a lamp means to use it to produce light. A lamp burns out when the fuel is exhausted or (in the case of an Edison lamp) when the filament wears out and breaks.
In this quote the reference is to the failure of an electric lamp:
Across from the house, at an angle, the lamp in front of 41 Mariner made a loud pop and went out. Joe glanced sideways and saw the brooding eyes of Taryn fixed in that direction. If the force of her emotion hadn't burned the lamp out, it certainly could have.
The author is saying that it almost looked like she broke the filament by staring at the lamp.
The phrase "burned of the lamp" refers to a burn from touching a lamp while it is hot. Here is an example:
George Armstrong, while descending the stairs in the rear of his little son, wife and lady guest, caught his heel on the step which threw him forward, coming in contact with those in front of him; they all landed in the hall promiscuously. They escaped serious injuries, but were badly bruised and burned of the lamp which was in the hands of one of the party.
--The Argus Reflector, January 18, 1883, page 5
The expression "burned of a lamp" is archaic. Today we would say "burned by a lamp".
Perhaps you meant "burning of the lamp" which refers to its operation. You could say:
The burning of the lamp produced a hissing noise and an unpleasant odor.