I have yet to see an oil injected engine that doesn't gum up after storage, especially if your 340 still has the original B38 butterfly carb!
If the oil system isn't flushed at the end of a season it will certainly gum up. Normally, smoke is a good thing (blue smoke anyway) that may just be extra oil being injected. Piston burn comes from a lean air/fuel condition. With 2 stroke oil slightly raising the octane rating of the fuel used, I have seen those 440s burn pistons from clogged oilers, though usually just before bearing / cylinder damage occurs from lack of oiling!
Make sure you clean the carb and service the oil system yearly & you shouldn't have any problems.
"They that can give up essential liberty to obtain a little temporary safety deserve neither safety nor liberty" - Ben Franklin
2012-2013 miles - 68 (Not bad for Indiana!!)