Interesting question Cassius.
There are not a lot of things which you can do yourself. Decoration, tiling (also external paving), putting lining walls in lower rooms is about it. Not even all works which can be done entirely without notification to the comune can be done entirely legally yourself.
The snag is that there are a lot of laws covering 'building work' and they are based on one or more considerations. Health and safety is one consideration, (here doing it yourself isn't so much of a problem, but hiring someone who hasn't got a DURC does become your problem).
Then there are laws which insist that the work is done by a sufficiently skilled 'abilitated' person. This clearly covers electrical and central heating work which needs to be certified, but there are also more minor plumbing works (changing bathroom fittings for example) which are controlled by idrosanitarie regulations, and you're not even supposed to do these.
Any work which needs a SCIA, or a Super Dia, or a full concessione needs to be done by a contractor with a current DURC. This is a way of checking the tax and national insurance status of the contractor - and it's basically to get the government money. Just having a DURC doesn't mean the contractor is 'abilitated' to do the work.
But clearly a lot of Italians 'do it themselves'. It's a matter of common sense, and knowing what you can do 'invisibly', and this depends on where your house is located, whether you have nasty neighbours, and whether you need a big pile of sand outside.
Here's an eight page legal opinion on 'doing it yourself' - it looks pretty solid to me, and maybe google translate can manage it.http://www.studioredaelli.it/database/COMMITTENTE%20IMPRENDITORE%20FAI%20DA%20TE.pdf