Just spotted this thread - it's already late, but there is a clear answer.
Slimcoin (see
here for more info) is currently the only cryptocurrency of my knowledge that is using Proof of burn in a "perpetual" way, like you say.
All participants can burn coins and if they are online "minting" with their wallet, they compete for block rewards. The more coins they burn, the higher the probability to find a block. The "power" of the burnt coins decreases over time but after a year less then half of it is consumed.
The block rewards are based on difficulty, so it's not easy to calculate an "exchange rate" - the more coins that all participants burn, the more difficult it becomes to get rewards.
For more info, you can read the
Whitepaper, visit the thread linked above or the
Slimcoin Project homepage.