Is maximum edge-weighted triangle-free graph NP-hard?
The problem is $NP$-complete even in the unweighted case (e.g.: all edge weights are equal to $1$). Instead of looking for a maximum edge-induced, triangle-free subgraph we may equivalently consider the minimum number of edges to delete from the original graph in order to leave a triangle-free graph. This problem (and other, related problems) are demonstrated to be $NP$-complete in [1]:

If $\pi $ is a property on graphs or digraphs, the edge-deletion problem can be stated as follows: find the minimum number of edges whose deletion results in a subgraph (or subdigraph) satisfying property $\pi $. Several well-studied graph problems can be formulated as edge-deletion problems.
In this paper we show that the edge-deletion problem is NP-complete for the following properties: (1) without cycles of specified length l, or of any length $ \leqq l$, (2) connected and degree-constrained, (3) outerplanar, (4) transitive digraph, (5) line-invertible, (6) bipartite, (7) transitively orientable. For problems (5), (6), (7) we determine the best possible bounds on the node-degrees for which the problems remain NP-complete.


Yannakakis, Mihalis. &quot;Edge-deletion problems.&quot; SIAM Journal on Computing 10.2 (1981): 297-309.