gabriel (2 inches taller)

First of all, thank you @xkeeper for this, who found this worked even after being told "It can't be done"

The Code

function love.draw()
  -- This is all you need to do in your Love2D Code, set the background to transparent.
  love.graphics.setBackgroundColor(0,0,0,0)
end

Setting up the OBS Capture

1. Game Capture
2. Mode: Capture Window
3. Allow Transparency: True/Checked

That's it?

That's it. As long as you don't change the background, anything drawn in the love2D window will be drawn as if on top of a transparent background. Great for fun effects without having to use a chroma key.