I think you are on the money with your burnt valve theory. The idea here when doing the "dollar bill test" is when there is a rattle like you describe, there is reversion happening at one or more of the cylinders. Reversion meaning instead of there being a clean exhaust flow (all one way), there is sucking happening from one of the exhaust valves during the intake cycle. This would indicate a burnt or non-sealing valve.
For the un-initiated, the "dollar bill test" is putting a bill (in the States it's a dollar bill or any denomination, actually) against the exhaust pipe while the engine is running. If everything is working well, the bill should only poof out from the exhaust pipe as the exhaust is only flowing one way ... OUT. If you have one or more burnt exhaust valves, the bill will rattle on the exhaust due to reversion happening. Reversion is where the exhaust is being sucked back up into the exhaust pipe during the intake cycle of the cylinder. This is really a very ingenious test and very old school in its origin.
To help with the diagnosis, you can perform compression and leak down tests on your cylinders. This will give you the health of your cylinders and will hopefully tell you which cylinder(s) is/are having issues.