All modern APIs (except some versions of OpenGL ES, I believe) on modern graphics hardware (the last 10 or so generations from ATi/AMD/nVidia and the last couple from Intel) support NP2 texture just fine. They've been in use, particularly for post-processing, for quite some time.
However, that's not to say they're as convenient as power-of-2 textures. One major case is memory packing; drivers can often pack textures into memory far better when they are powers of two. If you look at a texture with mipmaps, the base and all mips can be packed into an area 150% the original width and 100% the original height. It's also possible that certain texture sizes will line up memory pages with stride (texture row size, in bytes), which would provide an optimal memory access situation. NP2 makes this sort of optimization harder to perform, and so memory usage and addressing may be a hair less efficient. Whether you'll notice any effect is very much driver and application-dependent.
Offscreen effects are perhaps the most common usecase for NP2 textures, especially screen-sized textures. Almost every game on the market now that performs any kind of post-processing or deferred rendering has 1-15 offscreen buffers, many of which are the same size as the screen (for some effects, half or quarter-size are useful). These are generally well-supported, even with mipmaps.
Because NP2 textures are widely supported and almost a sure bet on desktops and consoles, using them should work just fine. If you're worried about platforms or hardware where they may not be supported, easy fallbacks include using the nearest power-of-2 size (may cause slightly lower quality, but will work) or dropping the effect entirely (with obvious consquences).