I have a fretless Fender J-Bass and I want to replace to flatwound strings that came with it with some roundwound strings. Would roundwounds damage the fingerboard because it has no frets?
Yes roundwounds will damage the fingerboard. You basically need to stick with flats unless you want to void the warranty and put poly onto the board. You might not notice the wear and tear right away but over time it will get serious.
Roundwounds will take chunks of your fretboard and bore grooves into it. The problem with this is that this will send your intonation out. Especially on the more commonly use frets.

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