Zero. LEDs don't have a heating element like other light bulbs. They also are the only light bulb that doesn't emit any UV light so they are totally safe for fabrics (no fading, no fiber breakdown).
There are only 2 potential downsides: (1) the cost, although that isn't as much of an issue when you consider the minimum life of these bulbs is around 10 years [some say 20, but that's probably not typically realistic], and (2) the size of the 19wt bulbs is like the size of a spotlight. They do fit within in my floor lamp shades, but just barely. However, since they emit no heat, they do not cause any melting/damage to the cheapy plastic shades that came with the lamp. You can get a smaller bulb, but then it obviously wouldn't be quite as bright.