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!
I think there are 2 main sources of confusion:
First, because of gravity, extending your arms feels like work. We're only interested in the radial movement, though, and in this direction, the skater's arms are pulled by the centrifugal force (in the long tradition of spherical cows in vacuum, we could replace the figure skater with two beads on a spinning rod).
Second, the idea of rotational energy as kinetic energy. The relevant work variable is (as already mentioned) the radial extension of the skater's arms, and as far as that's concerned, rotational energy plays the part of potential energy.
Think of the skater pulling in her arms as compressing a spring, and extending the arms as its release.
Going by either the bead or spring model, the rotational energy gets converted into kinetic energy of the arms, accelerated by the centrifugal force in direction of the radial work variable and ultimately dissipating via vibrations when the arms abruptly reach maximal extension.
Of course, if the skater doesn't let her arms be accelerated and slowly extends them instead, the energy dissipates right away, which might be the more realistic approach.
Best Answer
Normally, lifting someone up while stationary requires significant activation of the anterior deltoid (a relatively small muscle). When spinning, that force translates to back muscles, which are significantly larger/stronger, and your deltoids have to do less work.
An analogous situation is taking a weight and suspending it from a rope. Holding it stationary in front of you requires your bicep and anterior deltoid to activate, but if you start spinning yourself around, the centripetal force (+tension) keeps the weight elevated and now you just need to provide the centrifugal force inward to prevent it from flying away from you, which means pulling in a more horizontal direction, and so your bicep can work with your back rather than your shoulder.