I see it on both final render and in rendered viewport.

I think this may be a fundamental issue with texture interpolation. I've seen it in other 3D applications (such as Second Life), where if you have 1x1 repeats on a cube, you'll see some edges interpolated against the opposite side of the texture. What I usually do in those cases is set the repeats to 0.995 x 0.995. I tried that workaround on the example .blend file using the Vector->Mapping node, but it didn't work unfortunately.

Anyways, there are a couple workarounds you can use.

1. On the Image Texture node, go to the 2nd dropdown, and change it from Linear to Closest. Doing so will turn off smooth interpolation, so if you zoom in the pixels won't be smoothed, they'll be hard-edged. If you're only using this for big solid colors and aren't using any curved or angled shapes (much like the example image), this won't hurt anything.
2. Another possible workaround would be to use a higher-resolution texture, and perhaps putting a 1px thick alpha border around all sides of the image, so that there is no bleed-through.

At least one person with an nVidia card is unable to reproduce (https://developer.blender.org/T43795).

In T43795#291922, @Mike Pan (mpan3) wrote:

Cannot produce error either. Both render show the 1pixel border. GPU is a GTX970 running 347.09 Driver on Win7.

I experience texture offset using OpenCL on AMD hardware.

After playing with the transparency example and comparing the same result in the internal render, I'm inclined to say that this issue is more a lack of feature.
Exactly the same artifact appears in BI When using an Image or Movie type Texture, however under the Image Mapping section is the option Extension.
The setting Extend specifically repeats the edge pixels of the texture instead of wrapping around.
The setting Clip simply makes all pixels beyond the texture transparent.
The default setting Repeat wraps the texture, this seems the be the default behavior in cycles for any given image texture.
Using either the Extend or Clip settings immediately resolves the artifact.

Right now there is no way to change it, it seems. Having the option for Cycles Image Textures would solve this without work around's.
An interesting side note re: CPU / GPU texture offset, this example shows that the offset I am experiencing is down and left 1~ pixel, as the top line is slightly thicker and the left line is gone.

I was able to repro the issue on both CPU and GPU/CUDA.

System Information
64bit Manjaro Linux, kernel 3.14.33
CPU: AMD Phenom II 970 Deneb quad-core
GPU: nVidia GTX 650 Ti, driver 346.35

Please un-merge T43795: Texture Offset between Cycles CPU and GPU render as it is unrelated, thanks!

I'm not sure why you consider T43795 unrelated. The root of the issue is exactly the same -- wrapping coordinates on the image boundary when doing interpolation. The fact that some devices does expected interpolation in this case doesn't make any difference in the root of the issue.