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<blockquote>Initially, if I apply forces of 20 N and 15 N directly at the ends of a massless string (without any pulley or masses), the tension in the string is exactly 15 N, as expected.</blockquote>

No, it's not.  You've got a problem without a solution, like dividing by 0.  You can't apply 20 N to pull a string and then say the tension is 15 N.  It is 20 N by definition of applying 20 N.  Of course on the other end you have the same thing, resulting in 15 N.  That's an impossible situation for a massless object.  It simply can't be.

In the real world, there would be a net force of 5 N in one direction.  That divided by the mass of the object would tell you its acceleration.  In your case you have an impossible situation since the mass is 0.

<blockquote> why does the tension increase to 17 N when masses are used</blockquote>

Because of motion.  The 1.5 kg mass is being accelerated, so more than just its weight of 15 N must be applied to it.  This is of course assuming you are on earth where 1 kg weighs about 10 N.