I don't think they are that bad, but a picture of the whole bathroom would help. The chair rail looks like wood, which is odd and off-putting. I would replace the chair rail or at the very least, paint it the same color as the wall.

Both of our bathrooms were tiled this way...

I don’t think it’s that bad. Id still consider painting the chair rail. How much is your budget? If you DIY you could remove the tile and replace the Sheetrock underneath - that would be better than trying to get bead board to work on top of the tile.

Have you looked into painting the tile, or using those stick-on vinyl tile-cover decals?

I see nothing wrong with them. The probably stay a lot cleaner than having a painted wall there. They don't appear outdated to me.

My old farmhouse growing up as beadboard in the bathroom. I wouldn't recommend it. Even with "bathroom paint", it's just not a great environment for it. I think a fresh color of wall paint would be more effective and make the travertine look better. I don't consider travertine dated; I think it jsut depends on what's going on around it. You could also paint the chair rail the same color, if the white is too jarring.

Travertine is out of favor so I get that you want a change. But, they are in good shape and not too obtrusive.

I would get a new vanity and top. I would also do replace the mirror.

I agree with emil. The white trip is too jarring.

Don't cover or paint the tiles. Sorry to tell you but if you don't like them, remove and replace is the only good option. You could DIY to help the cost.

But to echo what everyone else said if you paint or get rid of the white chair rail, tweak the wall colors, and consider changing the vanities and mirrors I think you will find the tiles could fit in just fine.