You're talking about running a lawn mower on it... so I'm assuming you're using a column and making near azeotrope (90%+ ABV). If you're talking about a single run through a potstill or some similarly low ABV... don't do it. You'd probably be looking at multiple runs through a potstill before getting a product dry enough to put through an engine without messing it up. But if you got a column producing 90%+, small motors will run OK and if not they might only need a retune of the carburetor. I've dumped little jars of foreshots into my gas can and have had no problems in any small engines.
That being said, A couple of things to be careful of is that alcohol can burn hot... you can scorch cylinders and even melt rings if you're not careful. Ethanol can be very hard on engine seals and platics designed for sealing in petroleum fluids. Also, there's MUCH more water in home distilled spirit. Thus can lead to corrosion or rusting INSIDE your motor if you do it frequently enough. A better proposition is that you could cut it with gas to make something like a E25-E85 mix. Again, the water is an issue and you'll have to look out for phase separation because gasohol mixes can't really support more than about 0.5%-1.0% water in solution before the water begins to separate and settle out. And even if your still makes perfect azeo ethanol, it still has 4-5% water in it at a minimum. There is a host of information online about drying out that last few percent of water for using dried ethanol as biofuel. If you plan on using ethanol as a homemade fuel on a long term basis in any motor you care about, do some more reading. There are more than a few ways to get rid of that last 5% water in your spirit left over by typical distillation.
But like I've said, I dump the foreshots bottle into my gascan in the garage a few times and not noticed any problems. In moderation, I'd suspect that a bit of 180proof spirit mixed in probably cleans out the engine decently.
"Woe to those who are heroes at drinking wine and champions at mixing drinks" - God (Isaiah 5:22)
So evidently, God wants us to drink our whiskeys single barrel and our Bourbons neat.