Plan the run of the fence as far from the tree trunk as possible. Avoid routing the fence over any visible roots that are above the soil level, unless the fence doesn't come in contact with the ground at that point, such as is the case with a split rail fence.
Place string along the planned run, and mark the run with spray paint. Place short wooden stakes at every planned post hole.
Probe the soil with a metal rod measuring at least 4 feet long to discover if there is a main root under the planned post hole. If you do not hit any hard roots within the first 2 feet of soil, then it is safe to dig a post hole. If your probe hits a solid object, carefully dig around it to see whether it is a root or a rock. Re-cover the root with the soil, and adjust your fence post placement to miss the root by several feet. This may cause some fence panels to be either shorter or longer than others unless you adjust all the fence post locations at the same time.
Carry the supplies to the area near the tree instead of driving over the roots with a truck. This compacts the soil and may damage the underlying roots.
Consider stopping the fence and picking it back up on the other side of the tree if the fence line runs to close to the trunk. The gap between the fence can be filled in with a half-circle bench that allows you to enjoy sitting under the tree canopy.