"Do you want more pillows?"

Yes.

Basically, "fler" is used for things you can count (referring to the number of something), and "mer" is used for things you can't count (referring to the amount, or volume, of something). http://www.thelocal.se/blogs/theswedishteacher/2011/01/24/mer-mera-fler-or-flera/ a nice post explaining the difference. Some examples grabbed from there:

• Lars dricker mer kaffe än Bengt. (Lars drinks more coffee than Bengt.)
• Det finns fler sjöar i Finland än i Egypten. (There are more lakes in Sweden than in Egypt.)

Sometimes both are correct to use, depending on what you want to emphasize. In this case with the pillows, most Swedes would probably use "mer" and "fler" interchangeably.

Rule of thumb: use "flera" when you mean "many" or "several", otherwise either.

Jag drack flera flaskor öl = I drank many/several bottles of beer.

Jag drack fler flaskor öl = I drank more bottles of beer.

The more important (?) question is when you should use fler/flera vs. mer/mera, but I'm not sure where in the tree that comes in :)

In this context, and for the most part, you can use either.