At least there are solutions, found by trial and error and given as { x, y, z }:

sol = {{-3, 5, 2}, {-3, 5, 3}, {-2, 4, 2}, {-2, 5, 2}, {-2, 5, 
   3}, {-1, 4, 2}, {-1, 4, 3}, {-1, 5, 2}, {-1, 5, 3}, {-1, 5, 4}, {0,
    3, 2}, {0, 4, 2}, {0, 4, 3}, {0, 5, 2}, {0, 5, 3}, {0, 5, 4}, {1, 
   3, 2}, {1, 3, 3}, {1, 4, 2}, {1, 4, 3}, {1, 4, 4}, {1, 5, 2}, {1, 
   5, 3}, {1, 5, 4}, {1, 5, 5}, {2, 2, 2}, {2, 2, 3}, {2, 3, 2}, {2, 
   3, 3}, {2, 4, 2}, {2, 4, 3}, {2, 4, 4}, {2, 5, 2}, {2, 5, 3}, {2, 
   5, 4}, {2, 5, 5}, {3, 1, 2}, {3, 2, 2}, {3, 2, 3}, {3, 3, 2}, {3, 
   3, 3}, {3, 3, 4}, {3, 4, 2}, {3, 4, 3}, {3, 4, 4}, {3, 4, 5}, {3, 
   5, 2}, {3, 5, 3}, {3, 5, 4}, {3, 5, 5}, {4, 1, 2}, {4, 1, 3}, {4, 
   2, 2}, {4, 2, 3}, {4, 2, 4}, {4, 3, 2}, {4, 3, 3}, {4, 3, 4}, {4, 
   4, 2}, {4, 4, 3}, {4, 4, 4}, {4, 4, 5}, {4, 5, 2}, {4, 5, 3}, {4, 
   5, 4}, {4, 5, 5}, {5, 0, 2}, {5, 1, 2}, {5, 1, 3}, {5, 2, 2}, {5, 
   2, 3}, {5, 2, 4}, {5, 3, 2}, {5, 3, 3}, {5, 3, 4}, {5, 3, 5}, {5, 
   4, 2}, {5, 4, 3}, {5, 4, 4}, {5, 4, 5}, {5, 5, 2}, {5, 5, 3}, {5, 
   5, 4}, {5, 5, 5}}

(Log[2, 3 #1 + 5 #2 - 6 #3 + 3] > #3^2 - 9 #3 + 16) & @@@ sol

The solution in reals is simple enough:

Reduce[Log[2, 3 x + 5 y - 6 z + 3] > z^2 - 9 z + 16,
  y, Reals] // Simplify

y > 1/5 (-3 + 2^(16 - 9 z + z^2) - 3 x + 6 z)