What is the use of finding minimum number of straight lines to cover a set of points?
Although many papers in theoretical computer science claims practical applications for their work, this is unfortunately often simply not the case.  Usually, either the problems are too far away from being something useful (too simplified), or the algorithms are too far away from being practical (e.g. hiding big constants in the O-notation).

However, you can look at the papers


Approximation algorithms for hitting objects with straight lines (Hassin &amp; Megiddo) and
On the complexity of locating linear facilities in the plane (Megiddo &amp; Tamir).


They claim, e.g.


  The problem of hitting objects in the plane with a minimum number of straight lines has a military application.  In many cases when a bomber attempts to destroy
  targets on the ground, protected by anti-aircraft  missiles, it has to spend as little time as possible close to the targets. Thus, careful planning of an air raid on a multi-target site (for example, a cluster of fuel tanks) calls for a minimum number of times a bomber has to fly across the site. Moreover, each pass has to be carried out as fast as possible, hence for each dive into the site there exists a straight line (a "stick") along which targets are destroyed. Another application
  (in three dimensions) is in medicine where radiotherapy is administered
  by inserting a minimum number of radioactive needles into a certain area of the body so as to achieve a required level of radiation.


And also:


  For example, we may view the problems faced by a planner who has to locate r (linear) segments of a new railroad system so as to minimize the average cost to the users who have to reach the tracks from a number of different small communities. Thus, a straight line or a line segment is of natural importance in this context. Sometimes such problems are easier than those with point facilities. For example, it is much easier to find a line, so as to minimize the sum of distances to it from a set of given points, than to find a single point with the same objective.