Interesting sets of answers. But some still are a little misleading. I will try to sumarize.
1) Opening a file does not afect it in any way. Also closing it. Not in a viewer or editing program.
There is a chance you view the file diferent in diferent programs but that could be because how this program inteprets some information like color mode or color profile. But that process is only reading it.
There is a chance of small changes
2) Doing losless operations, like rotating an image. Normally the programs just re order the data of a jpg file, without analizing and recompressing. But I would not put my hands on the fire for all programs that supose to do that.
Small unnotorious changes
3) Opening and saving with the same compression on the same program.
A first recompression is done the first time you save a jpg file. If you save a second time the file with the same settings the original data loss is already done, but small changes can be applied again. Not in the same extent that the first one, but can be noticable doing this several times. But that depends on the program.
4) The most obvious is re saving with a different compression setting.
Not only on the "scale" on whatever the program has, but also the algorithm used. This is a little too technical but there are at least two main acompression algorithms 4:4:4 and 4:2:2.
You could use the "slider" on your program to top "quality", but if your program is using 4:2:2 and the original was on 4:4:4 you will have a significant data loss.
Here is a small paper I made some years ago so you can see what this data loss means, it is in spanish but you can use google translate: http://otake.com.mx/Apuntes/PruebasDeCompresion2/1-CompresionJpgProceso.htm
A total mess
5) If you open an image and re save it on a program that has limited capabilities. For example, a viwer could only save RGB files and not work properly with CMYK ones, or perhaps it does not understand the embeded color profile. You could totally ruin your image on saving.
6) Using a lot of compression. You save it for your website and compressed it. Do not delet your originals pease!
Only on the edited part of the image
7) The recompression is normally performed on all the image, but as I mentioned on point 3, it is not a lot of it if the image has not changed. When you edit an image this analisis has to be made again on this edited portion.
Remember that an editing can be categorized in three groups.
a) Color corrections, contrast, etc.
b) Altering one part of an image (red eyes, removing a person, cleaning unwanted spots)
c) A totally new collage.
So in some cases the image is a totally diferent one, at least for the analisis and recompression point of view.
In this post: https://photo.stackexchange.com/a/67434/37321 the user mentioned a program that does a very clever analisis of the existing compression and does not re compress it again if it is not necessary.