I have a demo app that I like, but I've deleted Xcode project (no backups) and now its provisioning profile date has expired and app doesn't work on iPhone. Is it possible to make it run again, maybe by renewing provisioning profile?
So, you need to renew the provisioning profile on the apple developer site. Once you do that, download the provisioning profile to your Mac and then load the profile onto the device either by emailing it to yourself on the device, using MDM (mobile device management), or using Xcode. No need to reinstall or re-sign the app unless the certificate expired.
If both the profile and the certificate expired, you will need to resign the app with the new cert and re-deploy it to the device.