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We study the limitations and fast-forwarding of quantum algorithms for linear ordinary differential equation (ODE) systems with a particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the ODE is inhomogeneous. On the one hand, for generic linear ODEs, by proving worst-case lower bounds, we show that quantum algorithms suffer from computational overheads due to two types of ``non-quantumness'': real part gap and non-normality of the coefficient matrix. We then show that homogeneous ODEs in the absence of both types of ``non-quantumness'' are equivalent to quantum dynamics, and reach the conclusion that quantum algorithms for quantum dynamics work best. To obtain these lower bounds, we propose a general framework for proving lower bounds on quantum algorithms that are amplifiers, meaning that they amplify the difference between a pair of input quantum states. On the other hand, we show how to fast-forward quantum algorithms for solving special classes of ODEs which leads to improved efficiency. More specifically, we obtain exponential improvements in both $T$ and the spectral norm of the coefficient matrix for inhomogeneous ODEs with efficiently implementable eigensystems, including various spatially discretized linear evolutionary partial differential equations. We give fast-forwarding algorithms that are conceptually different from existing ones in the sense that they neither require time discretization nor solving high-dimensional linear systems.