We report photon-noise limited performance of horn-coupled, aluminum lumped-element kinetic inductance detectors at millimeter wavelengths. The detectors are illuminated by a millimeter-wave source that uses an active multiplier chain to produce radiation between 140 and 160 GHz. We feed the multiplier with either amplified broadband noise or a continuous-wave tone from a microwave signal generator. We demonstrate that the detector response over a 40 dB range of source power is well-described by a simple model that considers the number of quasiparticles. The detector noise-equivalent power (NEP) is dominated by photon noise when the absorbed power is greater than approximately 1 pW, which corresponds to NEP ≈ 2 × 10^-17 W Hz^-1/2, referenced to absorbed power. At higher source power levels, we observe the relationships between noise and power expected from the photon statistics of the source signal: NEP ∝ P for broadband (chaotic) illumination and NEP ∝ P^1/2 for continuous-wave (coherent) illumination.