One potential danger of a burning smell emanating from a dishwasher is an electrical malfunction that might be a fire hazard. This is not necessarily the case, however, and care should be taken to unplug the unit and examine other potential causes before assuming the worst. Most dishwashers have plastic coated inner racks which are a good place to start looking for signs of a burn.
Plastic Rack Damage
Examine the plastic utensil holders and plastic-coated sliding racks for any signs of a burn. The plastic will typically appear blackened and may be pulled back, exposing the underlying metal. Carefully check each of the racks by removing them from the dishwasher and turning them over. If there are any burns, they are due to a malfunctioning heating element. Contact an authorized service center for your particular brand of dishwasher and have the unit examined.
Heating Coil Damage
The heating coil is the snake-like strip running around the bottom of your dishwasher. Plastic spoons or rubber container lids can often fall between the cracks in the dishwashing racks and burn on this coil. Ensure there are no pieces of plastic or other signs of a utensil having hit the coil. If plastic is present on the coil, remove it as best you can. Do not scrape the element with anything metallic.
Recalls for Fire Hazards
In order to be safe, ensure there are no scheduled recalls for your brand and model of dishwasher. Some models of dishwashers have had control panel issues that necessitated recall and repair. A simple online search for the model and make will typically turn up any manufacturer's recalls or known service issues.
If there are large amounts of plastic residue remaining on the heating coil, the smell of plastic may linger for some time. A good way to reduce the odor is to spread 2 cups of vinegar on the floor of the dishwasher before turning it on. Leave the unit empty to allow the vinegar to be dispersed freely on the walls of the unit and eliminate the odor.