Objective-C (and and its C++ counterpart, Objective-C++) can #include C (C or C++) header files as well as #import Objective-C header files. That pairs are even used is by convention.
In additon, ASP, Perl, Python, Pascal, and, since the file extension used for its includes doesn't have to follow a standard, PHP.
Edit: Most assembler source code also use "paired" included/implementation files by convention.
EDIT2 - ADDENDUM
Apparently there's some confusion as to the use cases that prompted the use of separate header files in C, and so some developers are dubious that certain other languages would also use header files to deal with those use cases in the same way.
- as the earliest C compilers ran on machines with a small amount of memory, it wasn't possible to load up symbol tables with all the source that would be incorporated into the final build. So each source file was run through multiple passes separately to generate the object code with necessary interfaces for other sources included in headers;
- as the earliest C compilers ran on machines with a small amount of memory, it was more efficient to distribute the shared elements of a module among more specialized headers which could be included selectively by sources rather than in one;
- as the earliest C compilers ran on slow machines, especially since they had a small amount of memory, it was time-consuming to compile lots of source files belonging to a module when only a few had been changed;
- sometimes source code maintainers don't share their code with outside groups, but instead make the object code available with headers of the required interfaces to aid in proper integration with linked code;
- and most importantly these days, programs and the language runtime sometimes deal with libraries, especially platform-dependent libraries, which are from an external environment or other language but which they can integrate with appropriate interface specifications in a header.
The last point is particularly true when a language supports a platform's GUI. I know it's been true on the Mac since its earliest days (when Pascal was its first high-level language) to now for
Perl (now automated), Python, and PHP. Its runtime environment may not make the inclusion process visible to you, but it's happening; especially if you use platform-specific symbols.
Creating bindings for C libraries in Free Pascal
h2xs builds a Perl extension from C header files, or use SWIG, or support C++
Interfacing Python to a C lib doesn't require just a header, but writing an extension!