In some places, the Swedish Lappland for example, it's perfectly fine and even recommended to drink water directly from streams.
Is it also recommended whilst hiking in the Alps, and if so is it only in certain areas?
In the core alps (Liechtenstein, Switzerland, and Austria), you can drink water almost everywhere directly without filtering. There are only two exceptions: If there is a thing or something like this that forbids drinking it, or if you can see an obvious reason not to drink it, like for example a strange smell or abnormal color.
In the other countries in the Alps (France, Italy, etc.) I would only drink water unfiltered if you can drink it directly from the source, i.e. glacial lakes or the source itself.
If you're on an extended trip away from civilization, I wouldn't recommend it anywhere except directly from a spring coming out of rock. If there are animals in the area, you can be sure they some have died, or done their business in the water and it could be contaminated. While water in the alps is likely safer that rivers or lakes in most areas, I wouldn't risk it.
My understanding is that anything above 4000 metres you can drink due to there being a low chance that anything living will affect the water i.e. animal faeces and bacteria etc.
I suppose you mean outside of village, so you are talking about streams and sources. In any alpine areas (France, Italy, Switzerland, Austria) I drink water unfiltered when I assume that there are no alps (place with cattle during summer) upstream, which worked for me. Of course you usually cannot be 100% sure about it, but almost so. If there is cattle upstream, you have a quite a chance of getting diarrhea. I did it twice until now when I was just too thirsty and once came to regret that decision :)
Many people attribute diarrhea to glacier water, which I cannot confirm. Unless you drink it constantly, the lack of minerals is not a problem either.
For dwells in towns in Switzerland you can drink always unless there is a sign telling you otherwise. Sometimes such a sign is present as the water is not officially tested, that also very often the case in alpine huts, while the water is fine. But you cannot know, so unless you have somebody trustful to advice you, keep away.
I'm Swiss and I've been drinking water from mountain streams all my life, without any altitude restriction, while observing three basic rules. You can drink the water if:
Drinking this water is one of my favourite things when hiking and I mostly go hiking carrying only a minimal amount of water to drink on the peaks where there are no rivers or sources. I hike often (in Ticino) and managed to survive for 44 years :-)
At a water faucet?
I would filter any untreated water in the Alps, and filter or boil melted snow.
Back in 1987 I bicycled over Fluela Pass in Switzerland. A young man chose to ride with me for a time and had me try water from a fresh Alpine stream. It tasted good and did not smell. More than 24 hours later I began to not feel well, feverish, tired, not hungry/but hungry. Four hours have having dinner I had gastrointestinal problems, every 20 minutes for about 3 hours. I wanted to die. It started as vomiting and ended up as diarrhea. I blamed my dinner; wonder if it was the water, after all?? It took days to get my strength back, but I continued biking....