Can ultra-low-res textures cause problems while rendering? This is a strange question, so comment if it's unclear.

I'm making a low-poly piece of art, where many of my objects will have only a single flat color (a cube that's all green, for example).

It seems that I should be able to use a very small texture for each object (the green cube could use just a single green pixel as its texture, for example).

I want to know if this will cause any problems. Should I use 16x16 or 32x32 textures, even though they provide more resolution than I actually need?

I am using the Blender internal render engine, though I also plan to export to Unity for a game project.
Yes, having a low res texture can cause issues, but they can be prevented. Here is an example:
This is the texture I will be using in the example, which is 16x16 pixels:

If we import this into Blender we notice that the pixels are all smoothed and the detail is completely lost. This is because the interpolation is set to Linear, which will fade from the color of one vertex to the next using a linear gradient:

Switching this to Closest brings back all the detail and makes the image remain sharp:

In your particular example, as long as this is enabled, you can go down to a 1 pixel size without any issues.