If images you are sent via iMessage on iPhone are being received as HEIC files (and in most cases it will not let you download it because the download button appears unresponsive) without showing the actual image itself, there is a simple fix that will save you from needing to reset your iPhone:
Settings -> General -> Transfer or Reset [Device] -> Reset -> Reset Network Settings.
This resets the network settings for your iPhone back to factory settings, which means you will need to enter wifi passwords again. It will restart your phone and then the problem will be fixed.