Q: Can a 12V tire inflator burn my car electrical system

Yes, it can be dangerous to plug high-current loads into the cigarette lighter! And, yes, this circuit should be fuse protected so that wiring or other components on the power path are not damaged. Not all manufacturers fuse-protect the cigarette lighter circuit however. They are designed for a prescribed load (the lighter coil). Other electronic devices (phone chargers, etc.) have much lower current requirements. Tire compressors however, can draw considerable power, and do so for an extended amount of time. There may also have been some other issues (corroded wiring connection, additional loads on that circuit, poorly made repairs) that caused an overload that was not fused. I would recommend having a electric problems inspection completed by a mobile, professional mechanic, such as one from YourMechanic, to diagnose this problem, get an accurate assessment of damage and cost estimate for repairs.