If the Spotify app installed on your iPhone doesn't recognize the songs you marked as disliked and fails to skip them when playing you might need to get the latest version of the app. This will make sure you get full functionalities and avoid issues with the app's features.
You can do this by performing a clean (more thorough) reinstall of the app on your iPhone. Check out this help page in order to find the steps for clean reinstall depending on the affected device.
In case this doesn't do the trick, would you mind sending us a video recording showing what you see on your end? Just attach the video to your next response to us. This way we can take a closer look at the issue. Make sure the recording doesn’t include any sensitive info such as payment or personal data.
Let us know how you get on! We'll be here in case you need any further help with this.