Please don't repeat the dripline myth - as a landscape architect I'm continually trying to to protect (and repair damage to) trees where the roots have been cut back to the dripline. The system I use is called the Critical Root Zone system (a utility standard in the UK) and is quite simple to work out:
Measure tree trunk diameter at 1.3metres (52 inches) above ground and multiply this figure by 29.5, the answer is the radius of protection you should allow for (this works for most tree types from narrow to broad canopies). If the tree has plenty of lawn space on one or more sides you might get away with cutting some roots, but where does the dominant wind blow from? because if you cut off all the roots on the windward side of a tree you risk it blowing over.
You should think about varying the spacing of fence posts in the vicinity of the tree, and digging the hole carefully by hand. Also what season is it where you live, autumn is the best time for this.
And finally you should consider the tree's monetary value - how much value is the tree adding to the total property value? Can you afford to lose this tree or will it chop $10,000 off your house price - just a few thoughts.