The problem is NP-hard, even in the unweighted case (all weights equal to $1$). There is a polynomial-time $O(\sqrt {\log n)}$-approximation algorithm due to Krivelevich, Nutov, Salavatipour, Verstraete, and Yuster, where $n$ is the number of vertices of the graph. This approximation ratio is essentially best possible (under a natural complexity class assumption). See this paper of Friggstad and Salavatipour for all the aforementioned results.

The problem is NP-hard, even in the unweighted case (all weights equal to $1$). Indeed, given a graph $G$ and an integer $k$, deciding if $G$ contains an Eulerian subgraph with at least $k$ edges is NP-complete. However, the problem is fixed parameter tractable (FPT) with respect to the parameter $k$. See this paper of Fomin and Golovach, and the references therein.