Any tips for using a scythe to mow a lawn?

Yes. Besides keeping your blade really sharp, when mowing dense bluegrass, you have to make sure that the blade gets down underneath the grass blades, otherwise the scythe blade can slide up over the grass, like it's on top of carpeting, and not cut much. To keep the blade low, you can actually press down on the lower grip, which will keep the blade pressed firmly to the ground. You may be cutting the grass pretty short the first couple of times, but the regrowth will be more tender, and easier to mow. You will have to mow less often with a scythe, than with a lawnmower.
  Also, grass is always much easier to mow very early in the morning. Don't worry, your neighbors won't complain, even if you're out mowing at 6AM on a Sunday morning. Mow when the dew is still on the grass, and before photosynthesis gets going. It will be much less work that way.

---

  Maintaining a lawn with a scythe, will have a different feel, look, and rhythm. Instead of a mechanically groomed lawn, you will have an inviting, natural looking meadow. If you eliminate the straight lines, and corners of your lawn, and fill them in with flowers, vegetables and perennials, and fruiting bushes and trees, you can have a permaculture Garden of Eden, instead of a mechanical/chemical lawn. Your lawn doesn't need to be layed out in straight efficient lines, if mown with a scythe. You can literally think outside the box" with your lawn design. 
 Always spread the windrows of cut grass out again, with a leaf or hay rake, if your going to leave the grass clippings on the lawn. Otherwise rake it up while it's still green, and add it to the compost, or dry it out on the lawn, and then rake it up and use it as mulch.  When I was first starting out, I used to practice making hay with my lawn grass for fun, and fed it to my girl's pet rabbits in the winter. The rabbits didn't like it all that much, but it gave me a feel for the hay curing process. (BTW rabbits like red clover and alfalfa hay the best.)
 

If you do try to make hay out of lawn cuttings, beware of dandelion stalks. Even if you mow when the dandelions are still yellow flowers, when drying in the sun, the flowers quickly ripen into seedheads. If you have a lot of dandelions in bloom, it might be best to rake up the fresh cut grass and to immediately put it in a compost pile to hopefully get cooked before the flowers can ripen. If you want to try and make hay of your lawn grass with dandelion stems in it, beware. Even when it seems that the grass is completely dry, the dandelion stems won't be, and if you put your finished hay away, the dandelion stalks will re-moisten the hay and make it moldy. It's like trying to dry cucumber slices with garden herbs, at the same time. If you don't have too many dandelions, you can pick out the stalks by hand, if you have a small yard.

---

Author Marcel Theroux wrote a nice little article Scything in Suburbia, about the pleasures of mowing a lawn with a scythe. Guess who the "Wisconsin farmer" that he mentions is?

Push-reel mowers and scythes can handle anything that a power-mower can handle, and more, while being non-polluting. A lawnmower manufactured before 2012, creates an amazing amount of pollution. Newer lawnmowers are much improved, but still pollute. http://www.peoplepoweredmachines.com/faq-environment.htm