It depends on the Win32 program. Clearly, GUI programs cannot run for obvious reasons. Console applications on the other hand may be able to run.
In 1998 Chris Jones wrote a DOS program called WinEM (previously DOSCon) which attempts to emulate the Win32 API so that Windows console programs can be run under DOS. According to the manual, he had apparently intended to support GUI programs and even Direct-X games but had, as of version 1.00, implemented only basic DX and windowing functions (ie, only enough for “'hello, world'-in-a-window”).
I cannot find any newer versions, so I suspect that it turned out to be more difficult than he hoped and so he abandoned the project. Also, because the program was written during the time of Windows 98, it may (probably?) does not support later versions of Windows (2000+).