I think you might be better to wait until first frost at least or a month after. Most temperate pruning is done in spring Feb-April. In the tropics it can vary from place to place and plant to plant. Of course If your in the southern hemisphere it's reversed.
Diversified Food forest maker . Fill every niche and you'll have less weeds (the weeds are the crop too). Fruit, greens, wild harvest, and nuts as staple. Food processing and preservation are key to self self-sufficiency. Never eat a plant without posetive identification and/or consulting an expert.
If the tree is in good shape then it's probably best to wait for cooler temps. If the branches in question are small new growth or damaged/diseased then you could probably cut them without too much trouble. As long as it's not a lot of pruning that needs to be done then the risk is minimal, but be aware that most diseases and pests will seek out the cut areas of your trees as a place to establish themselves.
I had to do some serious work on a few really badly damaged apple trees last year, so I waited for early spring so as not to encourage any more problems than I already had. That being said, I often cut little suckers and water sprouts throughout the season as they appear.
though dormant pruning is much more common, I almost exclusively summer prune. dormant pruning stimulates rank regrowth the following season, which I generally try to avoid, as it usually makes more pruning necessary.
summer pruning, on the other hand, decreases vigor somewhat, which is what I'm after when pruning established trees. this is especially true for fruit trees on dwarfing root stock that are getting a bit too big. summer pruning can cause sunburning if done when the sun is too intense or if too much is removed.
I think thinning some lower branches on a chestnut would likely be just fine this time of year, depending (of course) on local conditions, the age of the tree, and your goals.
If you live somewhere temperate, you risk frost injury to the tree if you prune this late in the year. You want to give the tree a chance to heal before frost. Around here, the 1st of August is about as late as is safe. Then once the tree goes dormant for the winter it's okay to resume pruning.