This isn't like traditional wood staining. Because of the paint a wood stain won't be able to penetrate. I have a few ideas but all of them will be difficult at best.
1- Apply a thick stain and use a graining tool to provide a wood grain pattern. This also requires painting the walls a neutral light tan color for a good background.
I've done this on a small scale with steel doors and it isn't easy getting a good look. Plus the doors were laying on saw horses not vertical like your walls. A good gel stain might work.
2- Strip off the paint. This could be a good idea but it will be difficult, time consuming and no guaranty of success. Before going all out strip a small area first to see how the paint comes off and how well the area will take stain.
3- Simply repaint will another color other than white. This is by far the easiest and cheapest option.
That's all I can think of right now.