We derive statistical-mechanical speed limits on dissipation from the classical, chaotic dynamics of many-particle systems. In one, the rate of irreversible entropy production in the environment is the maximum speed of a deterministic system out of equilibrium, S¯e/kB≥1/2Δt, and its inverse is the minimum time to execute the process, Δt≥kB/2S¯e. Starting with deterministic fluctuation theorems, we show there is a corresponding class of speed limits for physical observables measuring dissipation rates. For example, in many-particle systems interacting with a deterministic thermostat, there is a trade-off between the time to evolve between states and the heat flux, Q¯Δt≥kBT/2. These bounds constrain the relationship between dissipation and time during nonstationary processes, including transient excursions from steady states.