Harris, the Chiefs' No. 2 tight end behind Travis Kelce last season, will sit out the season opener against the Chargers on Sept. 9 in Los Angeles.
"We respect and support the league's decision in this matter and will have no further comment on the situation,'' the Chiefs said in a statement.
In March, Harris was sentenced to two days in jail and fined $500 after pleading guilty in Bates County (Missouri) Circuit Court to one count of marijuana possession and one count of unlawful possession of drug paraphernalia, both misdemeanors.
Harris was also given two years' probation and ordered to complete 80 hours of community service and an NFL drug program.
Harris, 26, has played four seasons with the Chiefs. Last season he set career highs for receptions (18) and yards (224).
He is eligible to participate in all offseason and training camp practices and preseason games.