The answer is yes.

A measure-preserving invertible shift $T: X \to X$ on a probability space $(X,\mu)$ is said to be [weakly mixing][1] if $\lim_{N \to \infty} \sum_{n=1}^N |\langle f \circ T^{-n}, g \rangle|^2 = 0$ for all $f,g$ in the Hilbert space $L^2(X)_0$ of mean zero square-integrable functions, and [strongly mixing][1] if $\lim_{n \to \infty} |\langle f \circ T^{-n}, g \rangle| = 0$ for all such functions.  There are examples of systems that are weakly mixing but not strongly mixing; see for instance [this previous MathOverflow post][2] for some examples (indeed in certain technical senses a "generic" shift is of this form).  Note that weakly mixing shifts have no eigenfunctions in $L^2(X)_0$ (indeed this is an if and only if, by the discrete version of the RAGE theorem).

Such systems $(X,\mu,T)$ are discrete flows, but they can be converted into continuous flows by the standard device of taking a [suspension][3].  Namely, let $\tilde X$ be $X \times {\bf R}/\sim$ where we quotient by the equivalence relation $(x,t) \sim (T^{-n} x, t+n)$ and endow this space with the product measure $\tilde \mu$ of $\mu$ and Lebesgue measure on the unit interval, and the continuous shift $\tilde T^t (x,s) := (x,s+t)$.  If one then lets $H \equiv L^2([0,1]; L^2(X)_0)$ be the Hilbert space of functions $f \in L^2(\tilde X)$ that are of mean zero on every time slice $X \times \{t\}$, and lets $U(t): H \to H$ be the [Koopman operator][4] $U(t) f(x,s) := f \circ \tilde T^{-t}(x,s) = f(x,s-t)$, one can easily verify that $U(t)$ is a strongly continuous unitary flow (and thus of the form $e^{itA}$ by [Stone's theorem][5]) that has no eigenfunctions, but such that $U(t)$ fails to weakly converge to zero (even if we restrict $t$ to the integers, in which case the continuous flow basically collapses back to the discrete flow $T^n$).


  [1]: https://en.wikipedia.org/wiki/Mixing_(mathematics)
  [2]: https://mathoverflow.net/questions/47080/examples-of-transformations-which-are-weak-mixing-but-not-strong-mixing
  [3]: https://en.wikipedia.org/wiki/Suspension_(topology)
  [4]: https://en.wikipedia.org/wiki/Composition_operator
  [5]: https://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups