If you've had heart disease in the past, have you gone up to Jungfrau or Schilthorn and what has your experience been? Of course, I will be contacting my cardiologist before travelling to Switzerland, but was curious to hear any real world experiences.
Airliners are typically pressurized to the equivalent of 6000-7000 feet of elevation.. Do you have issues in a plane?
Are you planning on climbing or just riding a conveyance of some sort? How high are you planning to go?
My husband has CHF and has visited both peaks with no ill effects. We would not stay overnight or exert ourselves too much, but a visit for a couple of hours has been safe.
Consult your doctor. Everyone's case is different.
I've been to both locations and had no serious problems at either one. However at the Jungfrau I became a bit short of breath after an hour or so and it was definitely more effort to get around, especially climbing stairs. The Sphinx Observatory there is one of the highest points at 11,716 ft but I was able to manage some time there.
I've been to Piz Gloria and the Schilthorn numerous times (9,744 ft.), and I've never had any problems there even if spending a few hours having a relaxing lunch at the revolving restaurant and exploring the James Bond exhibit.
One location where I had more serious problems was the Aguille du Midi, above Chamonix, France. It's at 12,605 ft. and I was able to manage a few hours there but experienced a sudden onset of shortness of breath and I had to get to the bottom as quickly as possible.
That's a wise idea to contact your cardiologist, as h/she will be the best one to provide accurate information based on your circumstances.
I had a quadruple bypass 3.5 years ago. Originally was planning on staying at Grindelwald as a base, and wanted to go up to Jungfrau and Schilthorn. Based on American Heart Association guidelines, both are considered high altitude and should be avoided for people with heart disease. Murren falls into the moderate altitude and I was planning on hiking there - but now I'm starting to rethink the plan. I mentioned it to my wife and she said that she would be worried the entire time that we went hiking in the Swiss Alps.
On the AHA site, it states, "Sudden cardiac death is the most frequent non-traumatic cause of death at altitude, according to the statement." That statement is giving me pause...
We'll see what my cardiologist says, but hiking and physical exertion activities may be greatly reduced.
You could stay in Lauterbrunnen, which is on the valley floor at an elevation of about 2500 feet. There are some nice hotels there, and it's the transportation hub for that area so you can easily travel to both sides of the valley. It would be easy to take a round trip to explore Mürren (5374 ft.) or Gimmelwald (~4475 ft.). You could start with the short cable car ride from Lauterbrunnen to Grütschalp and then connect with the BLM railway for the short trip to Mürren. When you've finished exploring there, either walk downhill on the paved track to Gimmelwald, or take the short cable car ride. After visiting Gimmelwald, take the cable car back to Stechelberg on the valley floor. You could either walk back to Lauterbrunnen or take the Post Bus.
If the cardiologist gives you the green light to visit the Schilthorn, the route to Mürren would be the same as described above. You could also take the Post Bus to Stechelberg and take the cable car from there. The rail station is at the opposite end of town from the cable car station so if you use the BLM routem a short walk would be required.
For a good view of hikes possible in that area, you might have a look at THIS book which has an excellent description of hikes in that area.