The requirements for Winelib are similar to those for Wine.
Basically if you can run Wine on your computer then you can run Winelib. But the converse is not true. You can also build Winelib and Winelib applications on platforms not supported by Wine, typically platforms with a non i386 processor. But this is still pretty much uncharted territory. It would be more reasonable to target one of the more mundane i386-based platforms first.
The main difference is that the compiler becomes much more important. It is highly recommended that you use gcc, g++, and the GNU binutils. The more recent your gcc compiler the better. For any serious amount of code you should not consider anything older than gcc 2.95.2. The latest gcc snapshots contain some useful bug fixes and much better support for anonymous structs and unions. This can help reduce the number of changes you have to do in your code but these are not stable releases of the compiler so you may not want to use them in production.