I've had my carseats in a few cars with non-removable headrests. The profile of the headrest determines whether it's a problem or not. I had my Radian in my Aunt's Audi that has headrests that stick out forward of the seat back. That resulted in a huge gap behind the carseat's back. It was a very poor install. In my Caravan (which does have removable headrests, but let's pretend it didn't for second...) the headrests are flush with the seat back and make hardly any difference at all when installing carseats. (I can slip the headrests on or off behind my installed Nautilus - it was a little tighter behind the Radian.)
In some GM vehicles, I could get the headrest to flip backwards (but not come off) if they were in the way. (We rented an Uplander that that worked in.)
Some carseats work better with problem headrests than others too. Some short-shelled seats can sneak in underneath the headrest.
I know there's one Jeep vehicle that's non-removable headrests are virtually incompatible with any carseats. :thumbsdown:
Too make a long story short: the compatibility of non-removable headrests with carseats depends on the headrests and the carseats.