DNA origami is a widely used method to construct nanostructures by self-assembling designed DNA strands. These structures are often used as "pegboards" for templated assembly of proteins, gold nanoparticles, aptamers, and other molecules, with applications ranging from therapeutics and diagnostics to plasmonics and photonics. Imaging these structures using atomic force microscopy (AFM) or transmission electron microscope (TEM) does not capture their full conformation ensemble as they only show their shape flattened on a surface. However, certain conformations of the nanostructure can position guest molecules into distances unaccounted for in their intended design, thus leading to spurious interactions between guest molecules that are designed to be separated. Here, we use molecular dynamics simulations to capture a conformational ensemble of two-dimensional (2D) DNA origami tiles and show that introducing single-stranded overhangs, which are typically used for functionalization of the origami with guest molecules, induces a curvature of the tile structure in the bulk. We show that the shape deformation is of entropic origin, with implications for the design of robust DNA origami breadboards as well as a potential approach to modulate structure shape by introducing overhangs. We then verify experimentally that the DNA overhangs introduce curvature into the DNA origami tiles under divalent as well as monovalent salt buffer conditions. We further experimentally verify that DNA origami functionalized with attached proteins also experiences such induced curvature. We provide the developed simulation code implementing the enhanced sampling to characterize the conformational space of DNA origami as open source software.