In my engineer's world of absolutes, the answer is; yes, round wound strings will chew up a fretless neck. The question you're really asking is "How badly, and how long will it take?" And the answer is, "it depends". In the real world, there are too many variables for a simple answer - unless you'll be happy with a simple "Yes they will"...
1 Strings. They're metal; the fingerboard is not. Unless you use tapewound strings. rubbing a metal string -
any metal string - across it will wear it out sooner or later. Since round wound strings are more like a file than flat wounds, they will wear it out a little sooner - all other things being equal...
2. Technique. As has been mentioned, sliding vs. bending, and a light touch with your "fretting" hand, will go a long way towards minimizing wear on the fingerboard. But it
will wear; just later, rather than sooner...
3. Fingerboard material. Some woods are harder than others (Duh..

), which is why ebony (hard) is more common on fretless instruments than, say, rosewood (not so hard). Synthetic materials - like ebanol - are pretty darn hard, and are becoming more popular, because they are both cheaper and CITES friendly. But, ebony and ebanol are still softer that steel, so.. see no.1 above. I have an old fretless Kramer aluminum neck around here with an ebanol board. It looks terrible, but is actually still very smooth. It's very obvious that a previous owned played it with round wounds, though... Epoxy coatings, although very hard, are still not quite as hard as the strings. It may take a long time, but the metal strings
will win in the end...
So bottom line is; yes, round wound wound strings will probably cause noticeable wear sooner than any other type of string. However, it's your responsibility to minimize that wear, through proper playing technique, and the appropriate maintenance at the proper time - which is
before the neck gets "chewed up"...
