When traveling and leaving an empty house for a long time - like a month or more - is it better to leave toilet covers/lids up or down? No animals or anything like that to worry about.
Was going to put this in GD, but decided against it. 
Trying and failing to think of a good reason to leave them up.
In a super-dry climate, closing the lid might reduce evaporative losses a bit, delaying the day on which enough water has evaporated from the bowl/trap to allow sewer gas to flow into the house. In the absence of any general draft, a humidity gradient from the surface of the water out into the greater bathroom environ will convey water away. The total mass flow rate of water vapor is a product of the steepness of the humidity gradient (change in relative humidity per unit distance) and the cross-sectional area of the flow. “Lid up” presents the maximum possible flow area; “lid down” reduces the flow area to the 1/2" tall open band around the edge of the seat/lid.
In our guest bathroom, the bathtub/shower hasn’t been used in years (COVID), but I frequently use the toilet. Occasionally when flushing the toilet I can hear the tub drain gurgle, and I know that water has been evaporating from the trap; I turn on the faucet to fill the trap again, and I’m good for several more months.
More importantly, I turn off the water to the house before we leave for vacation. (We’re on a well, so it’s easy: I simply turn off the circuit breaker to the well pump.)
That you know of.
Any vermin that enter the house via any other route will be happier if they can access that water. Lids down is better.
It’s not uncommon to find rats inside drain plumbing, especially in multi-family dwellings. If they emerge into your toilet the lid might persuade them to return from whence they came. if the lid is open you’ve left them a standing invitation to the rest of your house.
There’s no upside to lids up.
Far more important, as @Crafter_Man says, is to run off the water. And the electricity or gas supply to the water heater(s). That’s just in case they spring a leak while the water supply is off. You don’t want the heat feature running on a dry-ish tank. That would be bad.
Yup.
Toilet seat does not matter one bit.
When we were gone for six weeks we turned off the water, but I’m pretty sure we left the lid up. Plenty of water in the toilets when we returned.
Can you just split the difference and prop them all up at about a 45 degree angle?
It depends on if you are a man or a woman.
This seems to be a good summary of the general advice, Thanks.
Count me among those who see no advantage to lid up - ever, vacation or not. As I understand it, closing the toilet lid before flushing is an excellent way to prevent it acting as an aerosol and covering your bathroom in tiny particles of the toilet’s contents. Having done that, why open it again? A closed lid also prevents items falling into the bowl accidentally. All at the (extremely minor) ‘cost’ of having to lift the lid on next use.
Keeping the lid closed by default also obviates the classic male/female argument - the next user is required to lift the lid regardless of their mode of use. A simple bit of gender equality in action.
There’s no reason not to keep the lid down.
I’m Narcissus and I simply cannot approve this message.
Leave toilet cover up or down when traveling?
In a mobile home, campervan or trailer caravan I strongly recommend down.
Since burglars are five times as likely to be men compared to women, you should leave the seat up, as a courtesy.
Or as a trap; maybe they’ll fall in and drown rather than burgle.
This is why I asked the question. Unless you’ve cleaned your toilet immediately before leaving for your trip, there at least some splash/crud with germs on the rim or underside. With the lid open, air exposure could possibly prevent a petri dish situation.
There is a thread somewhere on SDMB about Mythbusters investigation of this issue. The results were inconclusive.
Here’s what I did many years before that. I lifted both the lid & the seat, put a sheet of newspaper over the entire bowl, closed both the lid & the seat, then flushed the toilet.
I waited for the tank to fill completely, then lifted both the lid & the seat. The newspaper was perfectly dry without a single water spot. You can try it yourself.
I concluded that a flushing toilet is not “acting as an aerosol and covering your bathroom in tiny particles of the toilet’s contents.” If open minded to anyone telling me why I’m wrong.
You doubt the existence of toilet plume aerosols?
There is a tremendous difference between macroscopic water drops you can easily see and microscopic aerosols you cannot possibly see even in large numbers.
Having said that, it’s not a priori obvious that a toilet lid with large gaps around the perimeter would be all that effective at trapping aerosols that aren’t ballistic, but instead simply drift about on the air currents until they bump into a surface and adhere to it.
@Chela’s site seems to say lid down reduces aerosol particle count, but makes the many fewer particles both bigger and more bacteria-laden.
Which lid setting is a net win for actual practical human health is not addressed in that cite.