dipole moments are possible in atomic bonds, not lone pairs. The only influence lone pairs have on dipole moments is that lone pairs can contribute to the geometry of the molecule. Like in water for example, where the lone pairs on the oxygen make the bent shape in stead of a linear shape (if water were linear it would not have a dipole moment, as it is bent, it does have a dipole moment).
Now in BrF3, the main focus is on the T-shape. The shape would mean that the two F atoms on the opposite ends of the T cancel out each others dipole vector, so the resulting vector will be along the stem of the T.
In the molecule, it is not a perfect T however. The angle between the F's is 86 degrees. This means that the resulting vector will still be in plane (along the stem of the T), but it will not just have a contribution from the F bound there, there will also be a resulting contribution from the other 2 F atoms (the ones on the ends of the T). Vector calculations will give the resulting contribution.