Pass in the Alps that can be crossed by mountain bike?
Get a printed map and look for the 'postal roads'...

There are 'postal roads' in Switzerland that are closed to normal cars. These go over some high passes that are just not wide enough for regular traffic. They are called 'postal roads' because only the post bus goes on them. What is amazing about them is the descents - you can ride 'TdF' style without really worrying too much about what might be around the corner... Well, you still have to worry about that but you don't have some stalker of a motorist pissing you off from behind or that obligation to skirt the side of the road.

As well as the postal routes there are some mountain roads that run one-way only, alternating which way the traffic goes. These are really good because you know that only the cars that you let go in front of you are ahead and there is nothing coming the other way.

Off the top of my head I cannot remember any of these glorious roads and I would have to phone a friend to remind me of where they were. I did a few going West-East in the German speaking parts of the country. For me as a cyclist they were what the Nurburgring is to a motorsport enthusiast - well, better than that, actually.

There are also the roads closed when the pass is not passable, that will be winter time for those, probably not what you are looking for.

Then there are those towns that you can only get to by cable car and train. These are fully tooled up to take bikes and no adverse weather stops a Swiss train from not running on time. They really do have a different way of getting around that you will only get your head round by taking some of these options.

Although surrounded by gigantic mountains without a definable pass, Zermatt is one of the few places that is of particular interest to a cyclist. It is accessible by road yet is car free. Cars have to stop five miles or so further down the road and thereafter it is buses, taxis and bikes.

Currently there are some financial implications to visiting Switzerland. The exchange rate and their 'safe haven' status means that your money does not go very far there. You might want to look into staying over in Youth hostels as they are all very clean, well presented and more affordable than those chalets that the skiers stay in.

I am not sure where to start online for finding the legendary post roads. I used a proper map last time rather than 'google maps'. On proper maps the post roads are identified clearly. They were well worth the effort and helped me avoid those hellish motorway-roads full of big lorries carrying really useful things like yoghurt half way across Europe.