Yes it is possible to have a landfill fire and when they do sometimes they burn for months. Depending on where the fire is and what types of waste the landfill accepts has a lot to do with the type of fire and difficulty putting it out. Many landfills restrict tires but many others gladly take them in for an extra dumping fee.
The landfill in Bismarck, North Dakota was the latest landfill to experience a fire. The landfill allows tires to be dumped here but was very vague about what was actually on fire. The thick black smoke could be seen for quite a distance. Based on what the landfill allows to be dumped there it was quite possible the smoke drifting over the area contained toxic chemicals. The fire supposedly was put out in a few hours. It was shown however other hotspots still smoldering with the potential to again burst into flames. This was not the first fire at the landfill as neighbors remember at least 2 others.
There are landfills across the country that have underground fires that have been burning for over 10 years and more. Landfill operators downplay the health impact from a fire at their facility. The reality is fumes definitely can cause health issues both short term and long term for the surrounding areas.