To be fair to the original poster, they are ASKING.
In many situations like this I would imagine there are quite a few people who wouldn't be clear on what is just for your complementary use, and what is complementary for you to use and keep if you should so wish.
What with all this talk of what people use their BA F bags for etc.
Also I've been in some weird hotels where (situation 1) there is a note in the room saying you are free to keep the bath robe, and other hotels where (situation 2) there is a sign saying the bath robe is the property of the hotel but you are free to keep 1 towel.!?!?
Now yes it may seem odd to want a pillow, but the original poster has a good reason.
I doubt they would think BA expects all the pillows to be taken, but not necessarily that it is prohibited. Remeber many airline magazines say 'your free copy to take away', but of course they only expect a tiny element to be removed.
The way I read the initial post, the question was:
"I know it would be incredibly unusual for people to take a pillow and that BA don't expect them to need continual replacing at a fast rate, but strictly speaking, are you allowed to take it? Because..."