I have booted a 286 diskless, using a boot ROM on a network card, sending an image of a ramdisk containing a DOS startup floppy disk. The 286 then ran programs from the ramdisk. This could be used to send programs like telnet, print spoolers, routers and other standalone utilities that you want to run on a diskless (and maybe headless and keyless) 286. Here are some ideas for what you could run.
The method could be generalised to other OSes that could run on a 286, e.g. ELKS. See the discussion below for what needs to be done.