There are several things that could be causing the burning smell in your AC system—some of them are indeed dangerous and cause for concern, while others are harmless. Most commonly, a burning smell in an AC system is caused by:
Issues like burning dust are harmless and will resolve themselves. Others like a clogged filter just require a quick fix (replacing the filter) to sort the issue out. Still others, such as an overheating motor or a broken capacitor are serious—but mostly pose a threat to the AC system itself, not you or your home. Then there are problems like burning electronics—which release toxic fumes, and can quickly lead to widespread house fires.