We'll be in the Rhone Alps area next June, and would love some advice about where to base ourselves for 4 nights after we leave Chamonix and before we cross the Italian border towards the Barolo area. I've been looking at Lac d'Annecy, but it occurs to me that Lac Leman (e.g. Yvoire or nearby) could also be a good base. Which would people suggest?

The Lac d'Annecy is exceptionally lovely, but you need to keep abreast of news about rainfall in the area, because the lake is actually drying up. People have actually been wading across parts of it near the city of Annecy. In any case, it looks quite disappointing at the moment in addition to being the start of an ecological catastrophe. We will need a very wet winter to return the level to normal and in June, you can't even count on enough snow to have melted to assist in the refill. Lac Léman does not present this problem, from what I know.

You might also want to check out the Lac du Bourget and the lovely town of Aix-les-Bains.

I would visit the greater Lake Geneva area BEFORE your stay at Chamonix, as Chamonix is along the itinerary from Geneva to Turin - Barolo.
Yvoire is a rather isolated place. Evian, Amphion, Thonon or Le Bouveret (southern side of Lake Geneva) resp. Nyon, Vevey, Montreux, Caux (Northern side of Lake Geneva) would offer more possibilities. But all depends on your interests.

I suppose you know that there is a lot to see along the itinerary from Chamonix to Barolo:
Courmayeur Mont Blanc,
National Park of Gran Paradiso,
medieval city, Roman sites and museums of Aosta,
the 12 Aosta Valley castles,
Breuil-Cervinia and it's gondola to Testa Grigia (3500m), just in front of Matterhorn (5 kms away),
Gressoney la Trinite and it's gondola to Salati Pass and Indren Glacier,
the former Italian capital Turin (european capital of chocolate) with royal and other palaces and tons of museums of all kinds.....

Not so familiar with Lake Annecy but am with Lake Geneva and from say a Montreux base you can do easy hops to a plethora of neat sights - by boat or train or of course by car but boat floats on the lake are neat.

As for Lac Léman, my favourite city is Lausanne on the Swiss side or Evian on the French side. Luckily, they are connected by a ferry.

I think Annecy would just be Annecy, right? While "Lac Leman" could be Geneva, Lausanne, Vevey, La Tour-de-Peilz,Montreux, Villeneuve...

Yes that is my impression too about Lake Annecy - certainly not the plethora of neat things like Lake Geneva - may have boats but not like Lake Geneva where they take you to so many neat places. Switzerland is however much more expensive in many way than France.

I think it really depends on what you want to do, and where your interests lie. Annecy and the Lac are pretty much a singular destination. If you are using it as a base to see other things, Lac Leman may fit the bill. Yvoire is reachable from Chamonix in an hour, if you have a car. Evian may be better suited, as it is larger. However, like the others, I suggest another destination like Vevey or Lausanne, and using the train and ferry system to explore, and then your car to take day trips (e.g., to Gruyeres). Montreux is another possibilty, but I would not base on the west side of the Lac (just my preference, I used to live in Lausanne and work on that side)

Don't really need a car for Lake Geneva - boats and trains and buses go everywhere - the Chocolate Train is a special tourist train that goes to Gruyeres - lovely old walled city with a cheese factory to tour and then onto Broc where the train stops right near the Nestle Callier Chocolate Factory for Willie Wonkaesque tours - train leaves from Montreux. For lots on trains and boats in that area check www.ricksteves.com and BETS-European Rail Experts.