Adding option MaxExtraConditions -> All to Solve helps:

Solve[Rationalize[
  yCO == (0.015 a + 0.05 b)/(a + b + c) && 
   yCO2 == (0.07*a)/(a + b + c) && yN2 == 1 - yCO - yCO2, 0], {a, b, 
  c}, MaxExtraConditions -> All]

 (* {{a -> ConditionalExpression[0, 
     yCO2 == 0 && yCO == 0 && yN2 == 1 && c != 0], 
   b -> ConditionalExpression[0, 
     yCO2 == 0 && yCO == 0 && yN2 == 1 && c != 0]}, {a -> 
    ConditionalExpression[0, 
     yCO2 == 0 && yCO + yN2 == 1 && yCO != 0 && b != 0], 
   c -> ConditionalExpression[(b - 20 b yCO)/(20 yCO), 
     yCO2 == 0 && yCO + yN2 == 1 && yCO != 0 && b != 0]}, {b -> 
    ConditionalExpression[-((3 a)/10), 
     yCO == 0 && yCO2 + yN2 == 1 && yCO2 != 0 && a != 0], 
   c -> ConditionalExpression[-((7 (-a + 10 a yCO2))/(100 yCO2)), 
     yCO == 0 && yCO2 + yN2 == 1 && yCO2 != 0 && a != 0]}, {b -> 
    ConditionalExpression[(a (14 yCO - 3 yCO2))/(10 yCO2), 
     yCO + yCO2 + yN2 == 1 && yCO2 != 0 && yCO != 0 && a yCO != 0], 
   c -> ConditionalExpression[(
     3 a - 200 a yCO + (a (14 yCO - 3 yCO2))/yCO2 - (
      20 a yCO (14 yCO - 3 yCO2))/yCO2)/(200 yCO), 
     yCO + yCO2 + yN2 == 1 && yCO2 != 0 && yCO != 0 && a yCO != 0]}}*)