Yes, it does.
My take is that, without nonlinearity, folding is missing.
One of the main mechanisms behind classical chaos is the so-called stretch and fold. It can be visualized as a blob of initial conditions being stretched and then folded over itself by the mapping: stretching leads to a divergence of close trajectories (the hallmark of chaos), while folding keeps them bounded (and dense).
A linear system may be able to produce stretching, but this alone corresponds to a trivial behavior, divergence. But, in the spaces we're considering, a linear system cannot produce folding:
[none of the] orbits of a linear operator in finite dimensions [...] are dense in the space
Why can't linear functions produce folding?
- The reason is that folding means that an interval is mapped on itself twice, and that requires a non-monotonic function (and linear functions are monotonic), such as the logistic map's parabola.
Notice, though, that topology can provide the folding that a linear transformation alone can't, see, e.g., Arnold's cat map, which is a linear map on a torus.
You ask about the condition 3, but we don't need to address it directly:
if $f$ has a dense set of periodic points and is transitive, then $f$ must have sensitive dependence on initial conditions.
Hence only the first two conditions of the definition of chaos need to be verified when showing that a particular function $f$ is chaotic.
As for differential equations, any linear ODE, $\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}$, can be solved. More pictorially, the understanding of maps can be somewhat transfered to flows by means of Poincaré maps and, especially, Poincaré recurrence plots. See, e.g., Carroll's A review of return maps for Rössler and the complex Lorenz or Crutchfield's slides from the lecture Example Dynamical Systems.
