Can I use a 3-way light bulb in my 60 watt lamp?
If your fixture is only rated for 60 watt bulbs, then as you seem to understand already, anything else could overheat and cause a fire.

The manufacture of the lamp rated it for 60 watts for a reason - material types, distance of bulb, etc.  Usually there is a safety margin built in, so is your lamp going to combust the second you put a 100 watt bulb in it? Probably not, but why risk it?  The only safe thing anyone should recommend is to follow the manufactures directions.