Is a spinning object heavier than a non-spinning object? Does spinning an object make it heavier? 
A real-world example: 
I was mowing the lawn in front of my house, a lawn that tends to have some steep inclines. I realized that the lawn-mower was easier to move, and much easier to pull up inclines when it was off than when it was running. The only difference between a lawn-mower that's running and one that is isn't is that the blade is rotating and there's some electricity running through it as it uses up gas. I guess the most significant of those would be the spinning blade.
Does the spinning blade make the lawn-mower heavier as a whole object, and would that mean that spinning objects are heavier than non-spinning objects?
 A: As in the answer of @Mark Eichenlaub the mass of the lawnmower won't increase! Of course the blades of the lawnmower can have a pull-effect in which they might aid your foreward-movement (in which degree this might help, I'm not sure ...).
Of course a rotating blade creates an angular momentum, so if you were to make a turn with your lawnmower you'll need some extra force to change this momentum. So I'd say that turning would be harder! Also if your lawn is uneven, your lawnmower might tilt which will of course also change your angular momentum, this might make it harder to go in a foreward direction!
So yes spinning can make an object heavier (in the sense that you'll need to apply a bigger force), but only if you'll work against it's angular momentum or try to change the direction of rotation! This is actually the working principle of a gyroscope.
A nice youtube-film about another real-life application (aircraft) is found here. A third real-life example is riding a bike ! Due to the spinning of the weels it's easy to stay in an upright position (or hard to fall), once you stop this is no longer the case!
A: No, a lawn mower is not heavier when spinning in any significant way. If it is harder to push, this is probably because of friction working against rotation of the wheels.
In theory, the lawn mower has slightly-higher mass when spinning according to $E = mc^2$. For a spinning lawn mower this is on the order of $10^{-16}$ the mower's weight, or $10^{-12}$ grams. The number comes out small because the blades are moving slowly compared to the speed of light. Additionally, the lawn mower in practice loses much more mass than this when the blades start spinning due to exhaust from the gas. 
A: during operation the blade in the mower has very high angular momentum. the blade actually spins on a plane parallel to the ground, so depending on cw or ccw spin, the angular momentum points downwards or upwards respectively. this makes the mower detectibly more resistant to changes in pitch(going up or down a hill) or roll (sideways tilt).
