LEDs poses least or no amount of fire risk, due to the way these are made and how they produce light.
A bulb can be a fire risk due to two reasons. Explosion and overheating.
Incandescent bulbs have a vacuum in them and CFLs contain a gas. But LEDs are made of solid parts which can not cause an explosion.
Secondly, LEDs utilize the electricity very efficiently and do not dissipate a lot of heat. This makes them relatively cooler than other types of bulbs.
It does not mean that they do not get hot. They get hot but the heat that is generated is dissipated through a heat sink which is located at the base of LED. Hence, the efficient design, and the ability to dissipate heat evenly ensure that the LEDs do not overheat, avoiding any kind of fire risk.
There is an article by David Lee on LinkedIn, which discuss this matter in further detail.