Intergalactic filaments form the foundation of the cosmic web that connect galaxies together, and provide an important reservoir of gas for galaxy growth and accretion. Here we present very high resolution two-dimensional simulations of the thermal and chemical evolution of such filaments, making use of a 32 species chemistry network that tracks the evolution of key molecules formed from hydrogen, oxygen, and carbon. We study the evolution of filaments over a wide range of parameters including the initial density, initial temperature, strength of the dissociating UV background, and metallicity. In low-redshift, $Z ≈ 0.1 Z_⊙ $ filaments, the evolution is determined completely by the initial cooling time. If this is sufficiently short, the center of the filament always collapses to form dense, cold core containing a substantial fraction of molecules. In high-redshift, $Z=10^{-3} Z_⊙$ filaments, the collapse proceeds much more slowly. This is due mostly to the lower initial temperatures, which leads to a much more modest increase in density before the atomic cooling limit is reached, making subsequent molecular cooling much less efficient. Finally, we study how the gravitational potential from a nearby dwarf galaxy affects the collapse of the filament and compare this to NGC 5253, a nearby starbusting dwarf galaxy thought to be fueled by the accretion of filament gas. In contrast to our fiducial case, a substantial density peak forms at the center of the potential. This peak evolves faster than the rest of the filament due to the increased rate at which chemical species form and cooling occur. We find that we achieve similar accretion rates as NGC 5253 but our two-dimensional simulations do not recover the formation of the giant molecular clouds that are seen in radio observations. ; Comment: 17 pages, 9 figures. ApJ in press