Adding my list as an example.
piwik_gone_files.txt

Another idea: What's wrong with replacing the old piwik folder with the latest piwik version (from https://piwik.org/download/)
and just keeping the config in config/config.ini.php?
(Serious question, I might be missing some important files, that might get lost this way)
refs #11132

lost files:

• tmp/ (should be regenerated, maybe even a good idea -> Add CLI command core:clear-tmp #6402)
• misc/geoip.dat (can also be redownloaded)
• plugins from marketplace

That might be a solution, too, but I manage all my installed applications through DirectAdmin/Installatron, which is the way my hosting provider works.
And yes: I do have shell access, but I still prefer to manage everything (or as much as possible) centralized. Also, I'd like to provide my hosting provider with a workable solution for his DA users who do not have shell access (or shell knowledge).

Seriously? Nobody?
There is not a single Piwik developer who could help me generate a list of files that should (or should not) be in my installation? This worries me a little...

How are you updating Piwik? The automatic process should remove all unneeded files. If not there might be insufficent access writes.
The solution suggest by @Findus23 should work. Using Git downloads to compare might not be the best solution, as Git does not include vendor libs.

If not there might be insufficent access writes.

I have Apache runing on Windows platform and the web-server have full access rights to filesystem, but Piwik cannot delete these files and file integrity check failed.

And I run Apache on Linux (CentOS), and my web server also has full access.

Could you please tell me which file/script does the unneeded-files-check for the automated update process? That should help me get my 2 Piwik installations back up and running, that have been down for a week now.

That should help me get my 2 Piwik installations back up and running, that have been down for a week now.

What is the error message you get? Maybe it doesn't even have to do with unneeded files and can be simply solved.
What happens if you use a fresh install (as I mentioned before)

The error I get this time is "Error: ViewDataTable ID 'table' is already in use!".
The solution should be to delete plugins/CoreVisualizations/Visualizations/HtmlTable/PivotBy.php, as a websearch tells me.

For my other installation I don't remember the error message, but the solution was to delete plugins/Goals/Reports/BaseGoal.php (I think). Which I did, and I now have my dashboard back, but no hits have been recorded for a week.

For previous update errors I've had to delete plugins/CorePluginsAdmin/Tasks.php (I think), or manually empty the cache.
I don't remember the exact errors before that, but I run into trouble with almost every Piwik update, and the solution is always to manually delete files.
And since search engines always know which file(s) to delete, I know I'm not the only one running into this problem.

I haven't tried a fresh install, but I guess things would work then, because there wouldn't be any left-behind files.

No answer is also an answer.
Apparently there is no way to generate a list of files that do or do not belong in my installation.
And apparently Piwik insists on including/requiring files it doesn't need.
After 3 weeks without statistics, I just switched statistics providers.
As far as I'm concerned, you may close this issue.

The system check in Piwik 3 should tell you which files need to be deleted. You can see it by going to Administration => System Check.