We present an analysis of the mid-infrared emission lines for a sample of 12 low-metallicity blue compact dwarf (BCD) galaxies based on high-resolution observations obtained with Infrared Spectrograph on board the Spitzer Space Telescope. We compare our sample with a local sample of typical starburst galaxies and active galactic nuclei (AGNs) to study the ionization field of starbursts over a broad range of physical parameters and examine its difference from the one produced by the AGN. The high-ionization line [O IV]25.89 μm is detected in most of the BCDs, starbursts, and AGNs in our sample. We propose a diagnostic diagram of the line ratios [O IV]25.89 μm/[S III]33.48 μm as a function of [Ne III]15.56 μm/[Ne II]12.81 μm which can be useful in identifying the principal excitation mechanism in a galaxy. Galaxies in this diagram split naturally into two branches. Classic AGNs as well as starburst galaxies with an AGN component populate the upper branch, with stronger AGNs displaying higher [Ne III]/[Ne II] ratios. BCDs and pure starbursts are located in the lower branch. We find that overall the placement of galaxies on this diagram correlates well with their corresponding locations in the log([N II]/Hα) versus log([O III]/Hβ) diagnostic diagram, which has been widely used in the optical. The two diagrams provide consistent classifications of the excitation mechanism in a galaxy. On the other hand, the diagram of [Ne III]15.56 μm/[Ne II]12.81 μm versus [S IV]10.51 μm/[S III]18.71 μm is not as efficient in separating AGNs from BCDs and pure starbursts. Our analysis demonstrates that BCDs in general do display higher [Ne III]/[Ne II] and [S IV]/[S III] line ratios than starbursts, with some reaching values even higher than those found at the centers of AGNs. Despite their hard radiation field though, no [Ne V]14.32 μm emission has been detected in the BCDs of our sample.