LaRoche would have been paid $10 million next season if he'd taken the option while Burnett was up for a $3.5 million payday.
Both were key players in the Nationals' run to a major-league best 98-64 record and National League East title last season. Washington was defeated by the St. Louis Cardinals in an NL Division series.
LaRoche, who turns 33 Tuesday, hit a personal best 33 home runs and had a second consecutive 100-RBI season. He hit .274 in 154 games and this week was awarded his first Gold Glove for his play at first.
Burnett, 30, was 1-2 with a 2.38 ERA in 70 appearances last season, when he was regarded as one of the best left-handed relievers in baseball.