The old house, (being) painted white, looks much better.

[sitifan]

The old house, being painted white, looks much better. (My bold.)
Source: Wu Bingzhong et al., Practical English Grammar, Volume 5, page 10.
Is "being" optional or obligatory?

 

[Annabel Lee]

The old house, being painted white, looks much better. (My bold.)
Source: Wu Bingzhong et al., Practical English Grammar, Volume 5, page 10.
Is "being" optional or obligatory?

It's obligatory if the intended meaning is that the old house looks much better and happens to be being painted white.

If the intended meaning is that the house looks better now that it has been painted white, change "being" to "having been":

The old house, having been painted white, looks much better.​

 

[tedmc]

I think having "being" in the sentence means the painting process was on-going when the sentence was written.

"Painted white" or "having been painted white" in the sentence mean the same thing - the painting was completed previously.

 

[PeterCW]

This all reminds me of the literal translations of Caesar's Gallic War that we had when I was learning Latin.

All the suggestions, while grammatically correct, sound unnatural to me. In a real conversation most native speakers would say something like "The old house looks much better painted white".