Update: This issue has now been solved.
I have been on leave for a month, and came back to a software update for Java 1.6 on my Mac. As usual, I installed it.
Unfortunately, the next time I opened OOo to put together a tutorial, I was met with this message:

It is a known issue with OOo. You can see more information about this at the issue 115180 webpage.
At this time, if you haven’t already, don’t install the latest JRE (Java) update from Apple until Issue 115180 is resolved.
If you have already (like me), there are a couple of options available to you.
Option 1 – Disable JRE
Disabling the JRE (Java Runtime Environment) in OOo might be an option for you if you don’t use the features that require Java. For a full list of these features, check out the Java and OpenOffice.org page.
In OpenOffice (quickly before this error kicks in, or hit the escape key until you don’t see it any more – I had to hit it 20-30 times), select the menu: OpenOffice >> Preferences.

Then click on ‘Java’ under ‘OpenOffice.org’ on the left panel, and then deselect (remove the tick) next to ‘Use a Java runtime environment’ on the right side.

Restart OpenOffice.org (not 100% necessary all the time, but I found it just a reassuring step).
Option 2 – Install NeoOffice
According to this forum, NeoOffice has fixed the JRE issues caused by the latest upgrade.
From the NeoOffice website:
NeoOffice is a full-featured set of office applications (including word processing, spreadsheet, and presentation programs) for Mac OS X. We have created an office suite that is adapted to the unique needs of Mac users by taking the features in Oracle’s OpenOffice.org office suite and adding improvements.
You can download and install NeoOffice from:
http://www.neooffice.org
Option 3 – Workaround implemented using Terminal **Unstable
This is not for the faint-hearted, and has mixed results of working or (worse) OpenOffice.org crashing. If you don’t know what the Terminal is, this is probably not the best thing to learn on.
The Issue # 115180 reports that the JavaVM can be picked up by OpenOffice.org if you enter the following lines in the Terminal.
cd /System/Library/Java/JavaVirtualMachines
sudo ln -s ../../Frameworks/JavaVM.framework/JavaVM JavaVM
After doing this, my OOo installation crashed, so I reversed it with:
sudo rm /System/Library/Java/JavaVirtualMachines/JavaVM
For me, I am now using NeoOffice, and as soon as there is a fix, I’ll post it here.
Updates
Some users have reported success with resetting their profiles. Please comment below if this works for you.