An emerging debate over plans to arm some George Washington University police officers on the D.C. campus near the White House shows enduring tensions over how best to protect college communities nationwide from gun violence. On one side are administrators who argue that they need every possible tool to prevent mass shootings and respond to fast-moving threats. On the other are students and neighbors who are often suspicious of police and contend that an armed force would not make them feel safer. “No guns for GWPD!” a crowd of students shouted Monday outside the residence of the university president, Mark S. Wrighton. More than 100 demonstrators — some with signs expressing views such as “Guns off my campus!” — had gathered at Kogan Plaza and marched to Wrighton’s house in the Foggy Bottom neighborhood.