Masking is the best solution, there’s no better way to prevent smoke residue. That being said…
What sorts of things are you cutting? If you’re going to finish them then prefinishing can make clean up much easier. Apply your finish as you would normally, then cut the wood and the smoke residue will generally wipe right off with alcohol or ammonia-based cleaners.
There are tricks to removing masking with less fiber tear-out as well:
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Try pulling the tape at a low angle. That means that you feel it up on a corner and then pull it almost flat across the surface. That will make it so that you’re pulling along the surface instead of pulling the fibers up away from the surface.
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Pull across the grain. Less likely to pick up fibers if you go across them.
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Again, prefinishing the wood. It will generally strengthen the birch fibers and the masking won’t pick them up. This will probably fail if you’re using oil finishing, as the masking will probably not adhere to the oily surface.
And lastly if you’re going to be removing masking from lots of little pieces, you might like the “tape sandwich” method:
Also, if you really can’t solve the tear-out issue and prefinishing is not an option… cleaning the smoke residue from the final piece can be done lots of ways… but my favorite is a quick sanding with ~300 grit sandpaper. A good sanding block is a must here.