Will Asurion charge me for my replacement phone if I've rooted it
In most (all?) cases, rooting voids your warranty.  I would assume that in jurisdictions with sane laws, physical damage clearly caused by the dropping would still be covered; they can't reasonably claim that the rooting caused the damage or would prevent them from fixing it.

That said, most jurisdictions don't have sane technology laws, and most carriers seem to jump at any chance to screw people over for money.  If they find out it's rooted, I would expect them to try to charge you the full cost.  So unroot / flash stock if you can.