I live in a gated community with pretty tight HOA rules, and I have never heard of this, ever. As far as I know, my HOA has no control over anything inside my house, as long as it doesn't end up outside.
I don't even get it, unless it is a gated community of patio homes, or attached homes, why would the HOA care if you torched your own house? (I dont' really believe that Christmas trees cause fires if they are properly tended, BTW.)
They grow those trees on tree farms specifically for that reason. It's like a vegetable garden, and completely renewed when the time is right. It's not like they go out into the woods and cut down a hundred trees for a tree lot. The occasional private citizen may cut their own off their own property, but most of them are grown and trimmed for sales.
All that being said, I have had an artificial tree for about 20 years-the same one. A real tree the size of what we need (12 feet at least) would cost us about $150/year. So we have our PITA put-it-together-branch-by-branch tree and I can put it up whenever I want to.