you are right the OpenGL needs to have a display connected.
If you want to convert a DP to VGA you need to use active DP to VGA dongles. You will find a list of qualified dongles on AMD Eyefinity Technology - Recommended Adapters
If you use a DP to DVI you will have a digital signal (DVI-D) that cannot be converted to VGA by a passive DVI to VGA adapter.
Thanks for the advice, I'll try to make a dummy plug with a DVI to VGA adapter and report back soon.
Alright, the Accel B101B-003B DP to VGA adapter has arrived. I've plugged a 100 Ohm resistor into ports 1 and 6 as per the How To Make a VGA Dummy Plug guide and now OpenGL is exposed when signing in via TightVNC instead of Remote Desktop. An important note though, I had to sign out, then plug this adapter in, then sign back in to make OpenGL appear (unlike with a real monitor, where I didn't have to sign in and out to make it work, it worked instantly).
I've cut the resistor legs and laid it sideways, then used some tape instead of soldering or glueing so that we can reuse this (not-too-cheap) adapter later if we ever need to. Now it looks like this. Thanks for the tip, chm.