I guess it is upsetting to learn this, but it really isn't much different from any other inorganic molecule-- salt, sodium bicarbonate, potassium chloride. Would we be upset to learn that the drug used for lethal injection was in our food? Actually, it is. Potassium chloride is a salt substitute many people put it on their food for seasoning and is used in food processing, I believe.
The human body produces a certain amount of ammonia daily directly and indirectly (bacteria in the gut) and excretes it via a process within the liver to produce urea which gets eliminated in the urine, among other ways (some is exhaled from the lungs). There are lots of other reasons not to eat meat that are more worrisome to me than ammonia used in food processing. I'm guessing here, but a hunk of beef jerky probably has more worrisome constituents (gram wise) than the amount of residual ammonia in a burger.
The cure for a fallacious argument is a better argument, not the suppression of ideas.
-- Carl Sagan