Why doesn't the herbicide 2,4-D damage lawn grass?
According to this link from Purdue, the seed of the answer is this:


  The herbicide is used to kill broadleaf weeds, which are dicots, while monocot grasses, such as sorghum and corn, are more resistant. That's because grasses inactivate 2,4-D inside the plant, while broadleaf dicots do not.


But on the other hand, Song 2014 has this to say about 2,4-D:


  However, the underlying molecular mechanism of how auxinic herbicides selectively kill
  dicots and spare monocots is not
  understood yet (Grossmann 2000; Kelley
  and Riechers 2007; McSteen 2010). The mechanisms of auxin biosynthesis, transport,
  and signal transduction
  are conserved in monocots and dicots make this question more complex (McSteen 2010). Early research has
  proposed that the selectivity of auxinic herbicide is because of
  either limited translocation or rapid degradation
  of exogenous auxin, altered vascular anatomy, or altered perception of auxin in monocots (Monaco et al. 2002;
  Kelley and Riechers 2007). Auxin transport is influenced by plant vascular
  systems (Mattsson et al. 1999;
  Scarpella et al. 2006). The difference in vascular tissue structure between dicots
  and monocots may contribute
  to the selectivity of auxinic herbicides. In monocot
  stems, the vascular tissues (the phloem and xylem) are
  scattered in bundles, and lack a vascular
  cambium; in dicot stems, the vascular tissues are form
  ed in rings and
  possess a cambium. 


So in other words, we don't know yet. It seems to be very strongly tied to auxin transport, but we do not have the full mechanism of action; since auxin is such an important plant hormone, and the pathway is therefore highly conserved (that is, the genes tend to be nearly identical on the parts of the sequence that affect the phenotype), the general conclusion is that the pathway itself is not affected, but an accessory pathway clears up auxin in levels excess of any normal amount in monocots.

Edit: Whoops, got the author's first/last names mixed up.