To fix this issue for the first time, I needed to follow the advice from the question Google Chrome blocking downloads from my site and go through verification process.
To fix an issue for all consequent file uploads, I needed to upload file into the server and try to download it with the latest version of Chrome. It should then automatically send file for analysis (or you should click "Send file for analysis" in the message box otherwise, if you are doing this for the first time). After ~15 hours (max 72 hours) file should be verified. In my case with regular exe updates, I put the file into publicly accessible directory on my server with any other name (app_verification.exe instead of app.exe) and wait till verification completes. As soon as it completes I rename app_verification.exe to app.exe and there is still no problem.
Also, ensure your file actually doesn't contain code that can be treaded like virus (using virustotal.com). Don't forget to digitally sign your exe with trusted certificate.