Do modern Terrain awareness and warning system (TAWS) have blind spots?
This answer is only applicable to civil airline type operations. Military systems do have significantly different requirements which I cannot address.

Quick Answer:

No, they don't have "blind spots" in the sense of areas with no data available. 

A TAWS system works differently: 

In the airline I've worked for, I was "exposed" to Honeywell Mark V EGPWS systems, but I assume that other systems are similar. These systems have a world-wide terrain database. That database and the algorithms take a normal terrain clearance into account as well as the performance of the aircraft.

"Holes" or "Corridors" are "cut" into this "terrain clearance carpet" to avoid nuisance warnings when an aircraft needs to descent below the normal margins. I.e. during landing and takeoff.

So, these aircraft can safely operate world-wide unless there are new mountains growing (which actually happens with volcanos) or large man-made obstacles are built.

As with all databases on board of an aircraft, they are updated on a regular basis.

Except for this link, I was not able to find an awful lot of information from manufacturers.