Gabai's solution of the Property R conjecture shows that a minimal genus Seifert surface of a knot, capped off in the 0-framed surgery along that knot, is of minimal genus in its homology class. In particular, it is incompressible in the 0-surgered manifold. On the other hand, there may be incompressible Seifert surfaces for the knot that are not of minimal genus. (For example many pretzel knots bound incompressible surfaces of arbitrarily high genus.) Presumably, there may be such a (non-minimal genus) incompressible surface that becomes compressible in the 0-surgered manifold. Does anyone know an example of this phenomenon?

(this was too long for a comment):

I don't know an answer, but I'll make a suggestion. If you have a knot with infinitely many Seifert surfaces, then there's a closed incompressible surface (it's a kind of limit measured lamination). http://www.ams.org/mathscinet-getitem?mr=2420023

If you perform 0-framed surgery, and the Seifert surfaces remain incompressible when capped off, then this closed surface should also be incompressible in the Dehn filled manifold. So my suggestion is to find a knot with a closed incompressible surface in its complement which compresses in 0-framed surgery. The pretzel knot examples don't seem to work. One possibility is to stick a handlebody with this property in $S^3$, such as http://www.ams.org/mathscinet-getitem?mr=2032111. Then you would also need to check that summing a minimal genus Seifert surface with the closed surface infinitely many times remains incompressible. I'm not sure how to do this part.