The Nationals also placed catcher Jesus Flores on the 60-day disabled list on Thursday. Flores had surgery to repair a torn labrum in his right shoulder on Wednesday.
The 37-year-old Burke is a career .282 hitter with three home runs and 38 RBIs in 184 games over seven seasons. He played in 13 games with Seattle this season.
Washington lost 4-2 at Philadelphia on Thursday night.