What happens to copper pours that are 'too narrow'?
It depends on the fab.  It's possible that a PCB fab would process these thin areas, but they will be fabed wider than you drew them.  Could a that cause a short between the copper pour and a pad?  Unfortunately, yes.  Removing these risky places to avoid spec abuse is a good idea.  Fortunately, this is not difficult to do in the layout software.

In Eagle, change the width property of the copper pour.  Below is an example with 2 copper pours.  They have identical geometries.  The only difference is the width property.  The upper pour has width=0.005 (dimensions in inches).  The lower pour has width=0.016.  Notice that the lower pour didn't "go" between the pads.  U1 is SOIC.