We establish improved complexity estimates of quantum algorithms for linear dissipative ordinary differential equations (ODEs) and show that the time dependence can be fast-forwarded to be sub-linear. Specifically, we show that a quantum algorithm based on truncated Dyson series can prepare history states of dissipative ODEs up to time (T) with cost (\widetilde{\mathcal{O}}(log(T) (log(1/\epsilon))^2 )), which is an exponential speedup over the best previous result. For final state preparation at time (T), we show that its complexity is (\widetilde{\mathcal{O}}(\sqrt{T} (log(1/\epsilon))^2 )), achieving a polynomial speedup in (T). We also analyze the complexity of simpler lower-order quantum algorithms, such as the forward Euler method and the trapezoidal rule, and find that even lower-order methods can still achieve (\widetilde{\mathcal{O}}(\sqrt{T})) cost with respect to time (T) for preparing final states of dissipative ODEs. As applications, we show that quantum algorithms can simulate dissipative non-Hermitian quantum dynamics and heat processes with fast-forwarded complexity sub-linear in time.