Furthermore is used to introduce a new argument.
Further states that something goes beyond what was already said.
These two are clearly related, but not the same, and not interchangeable.
Generally if you used further at the start of a statement where you should have used furthermore the reader will be left wondering "further to what?".
My guess is, it would be safe to change all "Further, ..." to "Furthermore, ...", but the opposite is not true?
No, they are not interchangeable in either direction.
Further to the above considerations...
*Furthermore to the above considerations...
Furthermore, our partners might object...
*Further, our partners might object...
Further Affiant Sayeth Not (a rather old-fashioned expression still found on affidavits and other legal documents, meaning the affiant has nothing else to add).
*Furthermore Affiant Sayeth Not.
In each of these, those marked with an asterisk are incorrect.
It's precisely because it's not safe to change all "Further,..." to "Furthermore,..." that you were advised to do so - in the case in question your colleague thought you meant furthermore when you used the completely different further.
That said, it's quite possible that all the other places where you have used further at the start of a sentence not only can, but should be changed to furthermore, because that's what you actually meant. Indeed, quite likely if you are asking this question.