I did some researching the last days and since no one else has answered this question, I'll do it.
During the winter, but also in spring or late autumn, passes are often closed for cars, because of the weather and road conditions. If you're lucky, you can catch a day when the pass is still closed for cars, but it is already possible to drive over it. This occasions are rare, but sometimes it is possible. To check the current conditions and status of passes in the Alps, you can check this site. It is in German, but the meaning is really obvious.
At least in Switzerland, there are a lot of signed mountain bike routes. These routes also climb over passes and if not using a separate way for bicycles, at least one lane on the main road is reserved for bicycles. For example this route crosses the Albula pass.
In some countries in the Alps, there are also a lot of roads that can only be used in one direction at one time. So it is not car free, but at least the traffic comes from only one direction. An example for that is the road to the Gigerwald lake.
But maybe the best option is to look for hiking treks that are also accessible by bike. On most Swiss hike treks, it is not forbidden to also use a bike. Choosing this option, it is very important to prepare the tour carefully, to be sure that the whole trek is really ride able. A famous example is the Pasubio passage, that is also part of some famous transalpian competitions.
Last but not least, there exist some old paths, built by the Romans, that are still visible today, or where a new road is built at the same place. These roads are less traveled and it might be a good choice for bicycles. A famous example is the Via Claudia Augusta.