Since the Gamecube had the ability to play GameBoy games via a hardware addon, it would make sense if Nintendo decided to do this again, but for the DS and the Wii right?
There are some technical issues that would get in the way of this ever being a possibility and the short answer to the question is, no, DS games CANNOT be played on the Wii and likely never will.
The main reason we will never see this happen is due to the fact that the DS requires a unique 2 screen setup, one of which is a touch sensitive touch screen. Replicating this on the Wii will not be possible without some sort of hardware controller (essentially a DS all on its own) in order to play the games. This would defeat the purpose a lot.
A cool alternative might be to use the touch control on the bottom DS screen and stream the video over to the TV wirelessly, which is something that has become quite common with other technology in modern times, such as smartphones. This will likely never work due to the fact that the DS has very low powered hardware and is not capable of performing such actions, which again brings us back to needing a special controller to be able to do this.
Overall it makes very little sense for Nintendo to ever implement functionality that allows DS games to be played on the Wii. The cost to develop would essentially mean it is a brand new console and it doesnt mean it will ever make any money. Its not a simple solution like it was for the Gamecube.
For now it isnt possible for us to have DS games on the big screen, but that doesn’t mean it isnt possible on other Nintendo consoles, say the Wii U.