[Discussion] Legally

If I make a non-profit pokemon game, will it be taken down/forced to be removed?
Have any other non-profit fan games been taken down?

So long as it remains Non-Profit, and you don't claim you made the assets yourself, you should be "fine" in terms of legality.
Speaking honestly though, it is still illegal, however Nintendo/Game Freak hasn't been seen making any moves against non-profit fan projects.

I'm not quite sure on the history of the legality when it comes to fangames, but my limited understanding of copyright leads me to believe that so long as yours isn't confused with the original, nor does it make money off their intellectual property, nor does it somehow damage their reputation or misrepresent The Pokemon Company, you may create Pokemon fangames.*

Obviously to have absolutely no ambiguity over that, I'd recommend instead creating your own characters and story with no ties to Pokemon at all. However, it's worth saying that I don't have any knowledge of a not-for-profit fangame ever facing legal ramifications.

*Darnit, Jim, I'm a designer, not a lawyer! I can't give a professional legal opinion.

As both Derxwna and Java's Missingno said, as long as the game is non-profit and does not damage the franchise, you should be in the clear.

The only fan-game I have ever heard of that was shut down by Nintendo was a Pokémon MMO, which gained too much popularity, to the point where Nintendo was worried it could damage sales of official games and was hence shut down.

Still, you will most probably never run into any real trouble.
The worst that might happen is that you receive a Cease-and-Desist letter, which is still just a heads-up warning, that unless you stop, then they will take legal action.
So long as you stop if you do receive a CnD letter, you are not in trouble, so don't worry about it!

As a guy whose non-profit pokemon fan game received a C&D, it's definitely possible.