The official druntime (for dmd) does not support OpenBSD. In order for everything to work properly with any OS with D, druntime must support it. It's possible that it would build for OpenBSD and work with some stuff (I don't know), but for it to really work properly, it needs to specifically support OpenBSD. There are a number of version blocks specific to OSes, and unless a version block for OpenBSD (or Posix) exists, then that functionality is missing, and won't work. If enough is missing, then nothing will work.
Now, druntime does vary on some level from compiler to compiler, as that's where the lower level, system-specific stuff goes (e.g. the C bindings to glibc and Win32). So, gdc and ldc have their own versions of druntime and may very well support OpenBSD. However, the official compiler, dmd, does not currently do so.
So, if you want to use OpenBSD with D, you're going to have to look at gdc and ldc. I believe that the gdc developers in particular have done some extra work to add additional OSes and architectures, since it's trying to get into gcc-proper, but I really don't know what exactly it supports at this point, and I have no idea what the current state of ldc is. However, there's a decent chance that they support OpenBSD.