While it is always recommended that you stick to power of two textures, whether or not it is required is dependent on platform implementation and content pipeline features. OpenGL itself does not require the textures to be square, or powers of two.
See this link for more specifics on OpenGL Textures.
To add in some of the details from the comments, there are performance and portability issues that affect your choice of texture dimensions. While it is indeed often not necessary for your artists to produce textures in powers of two, placing this restriction on them can provide benefits over the lifetime of the project.
This is something that could be overlooked, especially with the latest generation of gaming consoles and high end PC hardware, but the topic of performance is always relevant in gaming.
Power of two texture dimensions allow for assumptions to be made with various post processing and computer vision applications. This becomes even more true when your textures become square, though this is not always an option for various reasons. Whenever you can place constraints that enable your engineers to make assumptions, you can likely facilitate performance games.
Another performance gain can be made in the realm of memory management when building a texture atlas, or a simple decal/sprite sheet. When your textures are composed of random dimensions, it becomes increasingly difficult to pack them together, and you will require additional texture atlas'. While this may sound insignificant if you are developing for the PC, or the current generation of consoles, hand held devices such as the 3DS, the Vita, and the world of mobile devices are very concerned with texture size and memory footprint.
Finally, compression algorithms are always best when they are applied to a very specific type of data, and textures are no exception. These techniques can not only be better at compressing, but also faster at it. Many hardware platforms today have support for texture decompression as a part of the graphics pipeline. This is best supported via very specific texture formats.
A more overlooked benefit would be platform portability. Not all platforms support the same texture formats, and while you can guarantee that any platform will support powers of two, not all platforms support random dimensions. This leads to the logical conclusion that producing all your texture assets in power of two dimensions increases the cross platform support that your project has. This also might sound insignificant, but we are seeing more big name games being ported to these limited platforms. Such a project the designers likely thought was as far fetched as a trip to the moon once was.