CFLs produce a lot more light for a given wattage input, or put another way they produce much less heat for a given light output.
If a fixture is listed for a 60 watt incandescent, then the FIXTURE should not come to any harm from a 60 watt CFL, since the heat output is similar.
It is unlikely though that the LAMP would survive for long in an enclosed fixture since CFLs are killed by heat.
In practice though a 60 watt CFL is a very large one with a light output similar to 250 watts incandescent, it wont fit in any common fixture intended for 60 watt incandescents.
For enclosed fixtures I find that a CFL of a wattage not exceeding about one third of the incandescent wattage to be a good rule of thumb.
Fixture listed for up to 60 watt incandescent, use a CFL up to 20 watts, which will still be brighter than the incandescent.
For fixtures that are open at the top, with only the lower part of the bulb enclosed by a shade, then there should be no heat build up, and a CFL may be the same wattage as the incandescent. In practice though a 60 watt CFL would look daft in most fittings intended for 60 watt incandescent, nothing unsafe if well ventilated.
Inferior CFLs sometimes catch fire and could ignite surroundings, reputable ones are made of non-flammable materials and incoroporate a thermal fuse.