This has a lot of problems, one of them being that any reasonable fee right now (when funds are needed) would raise very little. Another is that fees are so small this would create a lot of dust and therefore destroy a lot of the value, since the inherent cost in handling dust is high. To avoid this we would have to greatly increase the block time to ensure a large number of transactions per block.
It has been proposed to me privately that we could issue a crypto asset backed by a portion of transaction fees. That more potentially addresses the first issue but not the second.