The side in favor of a voting representative state that the district is under "taxation without representation" while those opposed use the constitution to show a voting representative would be unconstitutional (only states are allowed voting representation under the constitution).
What is really behind the battle though is geopolitics. The District of Columbia (really Washington since its only one city) is rock-solid Democrat. The bill supporters introduced legislation that would give Republican Utah an extra voting seat as well, in an effort to appease Republican opponents. However, because of demographic shifts Utah will get an extra seat anyways after redistricting in 2010.