In a post on Reddit, it appears there is a way to play original Xbox games on the Xbox One using XLink Kai.
According to their website, XLink Kai bills itself as a "global gaming network - bringing together XBox, Playstation 2 / 3, Gamecube and PSP users, in one integrated community." XLink Kai allows you to play Xbox games by using software running on your WIndows PC, Macintosh, or Linux that allows you to play system-link enabled games online for free.
As noted by a Redditor, Microsoft previously made it impossible to use Xlink Kai by "adding a 30 ms ping limit to the system link games on Xbox 360." Microsoft's move effectively stopped game play on tunneling services, including XLink Kai. At the time, the only way to work around the ping limit was to do so on a modded Xbox 360.
In the comments section, one Redditor posted a YouTube video showing the service being used to play Halo 2. So, it looks like the XLink Kai method works for the time being. However, it is more than likely that Microsoft will put the kibosh on this work around and you will only be able to play original Xbox games online on Xbox Live, if Microsoft allows you the option to do so in the future.