BUNS is an acronym for "buried unsatisfied (polar atoms)".
BUNS are a problem for protein design, since they make the protein less likely to fold successfully. The unsatisfied polar atoms are likely to form hydrogen bonds with something, preventing the protein from reaching its designed shape.
As explained in Foldit Lab Report 6, puzzles may have a BUNS condition that penalizes buried unsatisfied polar atoms. The BUNS Objective in the Foldit blog delves into the underlying science in detail.
The BUNS condition was added at the suggestion of Foldit scientist neilpg628. The Poly-Proline Design puzzles feature helixes made entirely of proline (at least initially). The backbone polar atoms of these poly-proline helixes are difficult to satisfy in Foldit.