Some DLNA servers offer plugins to permit access to media from web sites. Serviio has a number of them, such as YouTube for example. However, be aware that web sites often make changes that alters or simply breaks the ability completely. Most sites don't want you to access their media from anything other than their site, mostly due to the advertisements they are paid to offer. As mentioned, casting from some other device, such as a computer or smart phone/tablet might be the simplest way to make it work.
Dan Roku Stick (3600), Ultra (4640), Ultra (4670), Premiere (3920), Insignia 720p Roku TV, Sharp 4K Roku TV, Nvidia Shield, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network.