As it is stated, this property does not hold: indeed consider the sequence $(u_n)_{n \in \mathbb{N}}$ defined for $t \in [0, T]$ by $$ u_n (t) = \sin (2n\pi t) v, $$ where $v \in V$ is a fixed vector. This sequence satisfies the assumptions, but not the conclusion.

Classical results on this topic involve an assumption of the type $(u_n')_{n \in\mathbb{N}}$ is bounded in $L^2 ([0, T], V')$ and imply the strong convergence (see for example Evans, Partial differential equations, 1998, section 5.9.2).

As it is stated, this property does not hold: indeed consider the sequence of functinos $(u_n)_{n \in \mathbb{N}}$ defined for $t \in [0, T]$ by $$ u_n (t) = \sin (2n\pi t) v, $$ where $v \in V$ is a fixed vector. The sequence converges clearly weakly to $0$ in $L^2 (0, T; V)$. Also since $L^2 (0, T)$ is dense in $L^1 (0, T)$ and the squence $(\sin 2n\pi t)$ is bounded in $L^\infty (0, T)$, the sequence $(\sin 2n\pi t)_{n \in \mathbb{N}}$ converges weakly-* in $L^\infty (0, T)$, and thus the sequence $(u_n)_{n \in \mathbb{N}}$ converges weakly-* in $L^\infty (0, T, H)$. This sequence satisfies thus the assumptions, but not the conclusion.

Classical results on this topic involve an assumption of the type $(u_n')_{n \in\mathbb{N}}$ is bounded in $L^2 ([0, T], V')$ and imply the strong convergence (see for example Evans, Partial differential equations, 1998, section 5.9.2).