I would have thought any modern system that has a full legacy compatible BIOS should work... (aka no i3/i5/i7 boards with UEFI). Remembering that MS-DOS strictly uses BIOS services to do stuff like access HDDs, keyboard, etc, it doesn't access hardware directly. (That also mean you can use USB keyboards as well as a good BIOS will emulate the AT controller for the USB device, and it'll work fine). About the only real requirement that I would consider is that the BIOS support IDE mode for the SATA connectors.
As for Win 3.1, that's a different ball game. As long as you stick to standard mode and not enhanced mode, it just uses the BIOS for all hardware related tasks. For video, as long as the video card support VESA 2.0 BIOS extension, you could use one of the many generic VESA drivers for WIn3.1, otherwise you're stuck with VGA res @ 16 colours. Keyboard and mouse will have to be PS/2 in this instance.
Is the control application 16bit or 32bit (and uses Win32s)?
Is the add-in card ISA, MCA or PCI? If it's ISA, then this is about the only board I could find without going embedded or custom:
http://www.ibase.com.tw/2009/mb945.htmL
Also, you could try using FreeDOS instead of MS-DOS, as it's tested on new systems and has a bucket load of features not found in MS-DOS (like full USB Mass storage support, FAT32 support, etc), but unsure how Win3.1 will go running on top of FreeDOS instead of MS-DOS. Something you could ask in their forums.
http://www.freedos.org/
PS. Remember, even the brand new i7-2600K starts in 16bit real mode for full legacy XT compatibility!