It's FAR more important to make sure the saw is tuned correctly before cutting with it than any "special" break in procedures. If the saw has mixture screws you may have to remove tabs, modify them, remove tamper proof items, etc to fine tune. This brings up some controversy about factory warranties and such, but for sure IF you run one lean from day one it's probably going to need warranty work more times than not vs the P/C lasting the life of the saw. When tuning ALWAYS er just a tad rich vs a tad lean, which can be difficult with some saws due to rev-limiting and timing-retard modules built into the coils. Although important that's a topic for another discussion.
Anyhow, make sure that it has fresh fuel and correct mix-ratio from day one. It hurts nothing to give the first tank an extra "shot" of oil but even that really isn't a deal breaker one way or the other.
Since the piston is coming to the top of the hole about 100 times a second in the cut break-in of the rings happens before you get thru your first cut in a decent size log.
What we really need to avoid new or otherwise is running lean in the cut. This does three things that can KILL the P/C. Higher than needed RPM's, not bringing in as much lubricant, and higher EGT's. I use no special "break-in" procedures here and have saws still in service dating back to the late 1970's. My 480CD for example has never been touched, not even a carb kit or fuel line replacement and is flawless. Since day one when I'm finished with them for the "season" (and I do this with all my saws) the tank is dumped while it's running and I run the carb dry pulling the choke just as the engine dies out. This gets as much of the fuel out of the saw as possible without taking it apart.
Next time you need it only takes a few extra pulls to bring it back to life, but you will never have to worry about it being all gummed up with stale fuel and the "apple jelly" this new fuel develops in a vented system.......Cliff