At what alcohol level does alcohol start to contribute sweetness?

Alcohol in itself is not sweet, like sugary sweet. But alcohol definitely contributes to the body of a beer, like residual sugars do. In my opinion this is the main factor higher alcohol beer are experienced as sweeter besides the residual sugars.

Stronger Belgian styles like a tripel have relative low FGs (e.g. down to 1.008 or less) but a high alcohol content. This keeps the beer in balance and keeps it 'drinkable' or 'digestible' as the Belgians call it. On the other side there are British ales with low alcohol but high FG. Also balanced.

Note: The term balanced suggest other styles are 'out of balance' which sounds negative but this of course comes down to beer style/personal preference.

Sounds like you are on the right lines. Wikipedia on sweetness of wine says "How sweet a wine will taste is also controlled by factors such as the acidity and alcohol levels, the amount of tannin present, and whether the wine is sparkling or not. A sweet wine such as a Vouvray can actually taste dry due to the high level of acidity. A dry wine can taste sweet if the alcohol level is elevated." quoting pages 198-199 of Peynaud, Emile. The Taste of Wine: The Art and Science of Wine Appreciation. Introduced by Michael Broadbent. Translated by Michael Schuster. San Francisco: The Wine Appreciation Guild, 1987 as the source but I don't have that book. I would also be interested in hard numbers.

While I think the discussion so far has been good, I also have felt like alcohol contributes something akin to sweetness.

While I appreciate the argument that high ABV beers tend to have high FGs which should result in a sweeter flavor, I often feel that the alcohol is more involved than this, especially with Double/Triple IPAs where I don't so much notice sweetness as an alcoholic flavor, much like vodka.

My theory is that alcohol, especially once it's fairly perceptible (as you mentioned, it seems to be most notable in the 8%+ range), offsets bitterness in a similar way to sweetness. Maybe you're experiencing something similar to that?

Clean pure ethanol diluted to about 5% with distilled water does have some sweetness.

I don't think there's a direct relationship between ABV and perceived sweetness. Residual sugars contribute to the sweetness. One reason for this can be the use of low attenuative yeast strains either on purpose or otherwise. High gravity beers require yeasts that are tolerant of the high alcohol levels. Use of these alcohol tolerant strains can produce dry beers even at high levels of ABV, however. So, in short, I think the answer of sweetness is more attributed to the combination of attenuation and alcohol tolerance of the yeast and not ABV.