There's no easy answer to that question. Most lenses "resolve" just fine and easily reach above 100 line pair per millimeter (lppm) at a range of apertures in the central area. The lens isn't the only thing in the optical chain, though. We may have antialiasing filters above the sensor that attempt to remove high frequency information, apertures in the lens that cause diffraction, and the sensor itself, which introduces a whole range of additional factors (Bayer filtration, microlenses, photosite pitch, less than 100% fill, etc.).
I usually phrase things differently than "outresolve our lenses." First, I write about "declining returns." At some point, diffraction and other factors get us to the point where having more pixels is not as rewarding as it used to be or that you’d expect from simple arithmetic. I believe we hit that bar at 24mp on DX (I first made that comment in 2003 based upon calculations I did then, and it looks very much like I was close to the way things are going to work out). Beyond that, the gain might not be worth the effort or expense.
Let me be clear, though: more sampling (e.g. more pixels) is always good in at least one respect: it provides a clearer picture (pardon the pun) of what the real analog data would be. I'll always take more pixels. But the reason why this question is here in the lens section is that those extra pixels may not truly be delivering more clearly visible resolution once we get past a certain point (though they may increase edge acuity).
So a better question might be: will my DX lenses still look great at 24mp? The answer is that some will, some won't. We already saw clear differences at 16mp. The 18-200mm, for instance, which looked great with 6mp cameras and was still good with 12mp cameras, doesn't quite do as well as some other lenses on a 16mp camera. It doesn't manage nearly as well on a 24mp DX camera, either.
By that I mean this: the higher resolution sensors are showing us the strong and weak abilities of a lens more clearly. You can often see that in the corners of the frame when you pixel peep, as lens performance in almost every aspect tends to drop as you move away from the center.
But there's another aspect to this question. If you shot with both a 12mp and 24mp camera and printed at the same reasonable size (say 8x10"), even a "weak" lens might look slightly better in some aspects with the higher mp camera. It certainly shouldn't look worse in such a comparison. Thus, my comment about the 18-200mm being weak on the high pixel count sensors tends to only be evident if you're pushing print sizes upward. If you had 300 dpi in the print from a lower pixel count sensor shooting with the 18-200mm and you kept the print size the same, the print should look about the same (again, maybe a slight advantage to the higher pixel count camera due to improved edge acuity). On the other hand, if you now think you can print 2x larger because you've doubled the pixel count, you're going to be in for a surprise: you're going to see the lenses' performance pretty clearly.