What is the black goo from the Fifth Element?

It's never explained.

There are a few theories though:

1. It's blood. It's simply darker than normal to indicate an even more sinister being. When Zorg (Gary Oldman) is called by Mr. Shadow, he starts to bleed it, suggesting that the mere force of evil in Mr. Shadow is enough to cause it.

2. It's some type of residue, caused by contact with pure evil. This is supported by the fact General Staedert has the same black ooze trickle down his face when the "planet" attacked his ship.

3. It's liquid fear. This is really just a different name for the two options above. It manifests itself in the same way and is caused by the Great Evil.

Obviously, all three of these notions are based on the exact same premise, giving it different names. TVTropes deem it a psychic nosebleed, again suggesting it lies along the same lines as one of the options above.

Unfortunately, these are all fan theories (commonly accessible across the web). There doesn't appear to be any definitive answer on the ooze from Luc Besson, the director, or any of the cast. Ultimately, I think these explanations are the most likely. Call it whatever you want, it's a way of signifying the presence of the Great Evil.

In the original script, the black fluid oozes out of a video monitor rather than Zorg's head

Shadow pauses an instant. A black slimy liquid starts to ooze from the top of the screen. Zorg is feeling worse all the time.

SHADOW: Money is of no importance... I want the Stones!

The novelisation is similarly unhelpful, merely remarking that

... A black, slimy liquid began to ooze from the top of Zorg’s head.

Notably, there's also a mention of similar fluid on the walls of the Fifth Element's chamber in both the script and the novel, implying that it's unlikely to be blood.