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C# (.NET Core), 28+13=41 bytes

n=>n-(1<<(int)Math.Log(n,2))

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All credit for this answer goes to NieDzejkob; the mathematical approach is significantly better than binary string manipulation (below).

C# (.NET Core), 86+13=99 bytes

n=>{var t=Convert.ToString(n,2).Remove(0,1);return t.Length<1?0:Convert.ToInt32(t,2);}

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+13 bytes for using System;

###UnGolfed###

n=>{
 var t = Convert.ToString(n,2).Remove(0,1); 
 return t.Length < 1 ? 0
 : Convert.ToInt32(t,2);
}

Unfortunately Convert.ToInt32 throws an exception on "", instead of returning 0.