This question resurfaces occasionally so I'll sticky it next to the forum acronyms thread.
I haven't searched each & every thread containing the phrase, but I believe we should award credit to Fire5Soon for the first mention of dryer sheets in a Jan 2004 post
as a reference to frugality:
Originally Posted by fire5soon
Personally, I could live on less than $15k/year including the mortgage... but I'm to the extreme on many issues I'm afraid. I've considered trying to grow some of my own food to reduce grocery costs. I cut my dryer sheets in half to make the box last longer (although I could line dry my clothes...).
It's a frugality reference. Of course it's also been touted in other threads as a flag, a symbol of the forum, and an inside joke... subsequently added by the admins to poster ratings.