"Further, ..." versus "Furthermore, ..." at the beginning of a sentence (meaning the same thing)?

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.

“Further” is usually employed as an adjective that modifies a noun. For the most part, it is NOT used at the beginning of a sentence (e.g., “Without further delay.”). It is synonymous with “additional.” By contrast, “furthermore” is an adverb that doesn’t modify anything. It is used to support or expand a prior statement. It IS generally used at the beginning of a sentence (e.g., “Furthermore, they allow pets.”). It is synonymous with “in addition.”

"Further" and "furthermore" are not interchangeable if you solely look at the dictionary. The definition for "furthermore" clearly concentrates on this usage, which is in contrast to the definition for "further". Merriam-Webster states this definition of "further" as definition 2 and compares it with "moreover". I criticize the dictionaries' lack of examples on words such as this. For example, if you click the link for "moreover", you will be shown that "besides" is a synonym. As a result, non-native English writers get in the habit of writing "Besides, ..." by itself. "Further" evolved to mean furthermore by people writing: "Further to this argument..." and now there is a situation where people think "Further, ...." and "Besides, ..." sound natural and acceptable to everyone. But just by looking at the differences of opinion here, they clearly don't. And if they did, there would be no need to have the word "furthermore".