Burton was a curatorial fellow at the
New Museum in New York City in 2002. She served as associate director and a faculty member of the
Whitney Museum's Independent Study Program from 2008 to 2010. Her position at Bard College ended in a group show called "Anti-Establishment", featuring artists who, in her words, work with "novel collective relationships and emergent models of engaged citizenship." The
Emily Hall Tremaine Foundation awarded her a $150,000 grant in 2012 to support her exhibition "Slow Dance". She curated a 2011 exhibition on
Sherrie Levine at the Whitney Museum, and with curator
Anne Ellegood, she guest-curated a 2014 exhibition at the
Hammer Museum on critique and appropriation. She also curated exhibitions for artists
Simone Leigh and
Haim Steinbach. In 2013, Burton returned to the
New Museum to become director and curator of its education and public programs, succeeding . She was selected as a 2019 Center for Curatorial Leadership Fellowship. In late 2021, she was named executive director of the
Museum of Contemporary Art in Los Angeles. Burton was to lead the MOCA with museum director
Klaus Biesenbach, but she was named director two weeks later when Biesenbach quit the museum. In July 2025, it was announced that Burton would become director of the
Institute of Contemporary Art, Philadelphia in November. == Personal life ==