Transiting planets in nearby young clusters offer the opportunity to study the atmospheres and dynamics of planets during their formative years. To this end, we focused on K2-25b-a close-in (P = 3.48 days), Neptune-sized exoplanet orbiting a M4.5 dwarf in the 650 Myr Hyades cluster. We combined photometric observations of K2-25 covering a total of 44 transits and spanning >2 yr, drawn from a mix of space-based telescopes (Spitzer Space Telescope and K2) and ground-based facilities (Las Cumbres Observatory Global Telescope network and MEarth). The transit photometry spanned 0.6-4.5 m, which enabled our study of K2-25b's transmission spectrum. We combined and fit each data set at a common wavelength within a Markov Chain Monte Carlo framework, yielding consistent planet parameters. The resulting transit depths ruled out a solar-composition atmosphere for K2-25b for the range of expected planetary masses and equilibrium temperature at a >4 confidence level, and are consistent with a flat transmission spectrum. Mass constraints and transit observations at a finer grid of wavelengths (e.g., from the Hubble Space Telescope) are needed to make more definitive statements about the presence of clouds or an atmosphere of high mean molecular weight. Our precise measurements of K2-25b's transit duration also enabled new constraints on the eccentricity of K2-25's orbit. We find K2-25b's orbit to be eccentric (e > 0.20) for all reasonable stellar densities and independent of the observation wavelength or instrument. The high eccentricity is suggestive of a complex dynamical history and motivates future searches for additional planets or stellar companions.