I would not want to hit a thick patch of fallen chestnuts with much speed! I'd slow way down, maybe 3-7mph, stand up on the pedals, off the seat, weight slightly back, and brace for unexpected handling reactions. Skinny, high-pressure tires or clipped-in feet are also concerns, if you use those.
You might want to try intentionally running over one or two single chestnuts at a time, at slow speed, to see what it feels like, so you'd have some idea what to expect if you hit some by mistake. It's going to feel quite a bit like hitting a round rock of the same size, so be prepared!
The sharp spines on the husks might puncture some tires, but most commuter tires are tough enough to take it. Even so, I'd avoid them if possible just to avoid that chance of a flat. If you do get a flat, be sure to find the spine that made it through to the tube AND REMOVE IT FROM THE TIRE, completely, or it will probably poke through the tube again.
It would be nice if the route you are riding was swept of those nuts. At least a couple of guys on here have swept various spots themselves, and most cities have a service number or website form to request street maintenance including sweeping. (Those contacts should be--& probably are--in a sticky thread here, but I'm not finding it, offhand.)