Vacuum birefringence Many of the papers (e.g., this) dealing with nonlinear electrodynamics treat a theory's prediction of vacuum birefringence as undesirable, but don't explain why it would be undesirable.  For example: 

Starting with the most general non-linear theory derived from an arbitrary Lagrangian depending on two Lorentz invariants of Maxwell’s tensor, L(P,S), he discovered that among all such non-linear theories, the Born-Infeld electrodynamics is the only one ensuring the absence of birefringence, i.e. propagation along a single light-cone, and the absence of shock waves. In this respect the Born-Infeld theory is unique (except for another singular and unphysical Lagrangian L=P/S). A beautiful discussion of these properties can be found in I. Bialynicki-Birula’s paper

The bold italicized emphasis is mine.
Other papers (e.g., this) treat vacuum birefringence in extreme fields as expected and natural.  
My question: is vacuum birefringence expected and natural, and Born-Infeld electrodynamics therefore deficient?
 A: I think I've found a partial answer in this paper: “FASTER THAN LIGHT” PHOTONS IN GRAVITATIONAL FIELDS– CAUSALITY,  ANOMALIES AND HORIZON", along with numerous papers by Patricio Gaete.
Vacuum birefringence per se is not necessarily bad, as long as it does not allow superluminal motion. Vacuum birefringence results in at least two different light speeds, and therefore two different light cones. As long as both light cones correspond to light speeds less than or equal to c, it's okay. 
Vacuum birefringence per se does not necessarily occur in QED since it hasn't yet been detected experimentally.  Some folks are looking for evidence of vacuum birefringence in gravitational lensing. It might be possible to observe birefringence via light that has passed close to a magnetar, but it could be very difficult to separate out the effects of a magnetized, rapidly moving plasma in the vicinity of the magnetar.  Detection of an electric dipole moment of the electron would suggest that vacuum polarization occurs.
Vacuum birefringence appears to be an area of ongoing theoretical, experimental, and observational activity, with key questions still unanswered.  The Born-Infeld Lagrangian is used widely in nonlinear electrodynamics and has a very attractive simplicity.  In its basic form, the B-I Lagrangian does not exhibit birefringence; but some generalizations of the B-I Lagrangian do exhibit it.  
So, I think the best answer that can be given is that at this point in time the answer is not known.  If vacuum birefringence is ever actually observed, the B-I Lagrangian rightfully will be considered deficient.
