I write small header-only and static-inline-only libraries in C. Would this be a bad idea when applied to big libraries? Or is it likely that the running time will be faster with the header-only version? Well, without considering the obvious compilation time difference.
Yes, it is a bad idea -- especially when integrated with larger libraries.
The problem of inline functions' complexity generally increases as these libraries are included and visible to more translations and more complex header inclusion graphs -- which is quite common with larger projects. It becomes much more time consuming to build as translation counts and dependencies increase. The increase is not typically linear complexity.
There are reasons this flies in C++, but not in C. inline export semantics differ. In short, you will end up producing tons of copies of functions in C (as well as functions' variables). C++ deduplicates them. C does not.
Also, inlining isn't a silver bullet for speed. The approach will often increase your code size and executable size. Large functions can create slower code. Copies of programs/functions can also make your program slower. Larger binaries take more time to link and initialize (=launch). Smaller is usually better.
It's better to consider alternatives, such as Link Time Optimizations, Whole Program Optimizations, Library design, using C++ -- and to avoid C definitions in headers.
Also keep in mind that the compiler can eliminate dead code, and the linker can eliminate unused functions.