There are really only 3 viable methods for handling the deleting of data:
- Prompt a confirmation, and make it loud!
- Soft delete the data so it can easily be recoverable if needed (this is much more work to implement though)
- Have an "undo" function. This is mainly only applicable to document editing types of features, and again is a lot of extra work
Never just delete data from a single click. It's far too easy for it to accidentally happen.
With that in mind, let's look at your specific scenario...
You have 2 main use cases, and I will give my suggestion for both.
User has added new field, typed a value, and saved it to the database. In this case, you most definitely want to prompt for confirmation to ensure the user doesn't accidentally delete existing data.
User has added new field, typed a value, but not yet saved to the database. This is less important for you to need to prompt for confirmation, but still should be considered. I would say you need to make a judgement call based on the type of data your input fields accept. You need to weigh up "annoying the user with needless prompts" vs "annoying the user with loss of entered data".
In summary, I will suggest that you always prompt for delete if the input contains a value. If a good enough reason appears down the line to remove the prompt then wait for that reason, don't try to preempt it.
I won't provide examples but I think you will find almost every application that's worth using will have some way to either confirm deletion, or provides an easy recovery/undo feature.