It is a good idea to put the axle on jack stands, if possible.
But you probably won't suffer and negative side effects if you do leave weight on the tires - except maybe staining the concrete of your garage/driveway.
I had to replace the tires on my truck this year, due to weather checking and some pretty impressive sidewall cracking. They probably still had 25k miles worth of tread left.
But they were on a truck that lives outside 99.9% of the time, in full sun all winter and 80% of the day in summer. Daily driven, but sun eats at the tires every day.
I've had plenty of tires flat-spot from being parked. They work themselves round again in short order. (And I have never had long-term effects from such.)
Suspension may be something to consider.
The torsion axle under my pop up tent trailer is collapsing again (4th axle for this trailer). I have been wondering for a few years - especially after this last summer - if storing it with weight off the wheels might help with longevity of these axles.
The previous owners had the trailer overloaded for each suspension failure, but I have been very careful to not do so. I think the bigger issue for this particular trailer is that it has a dry weight of (from memory, so +/- a bit) 1,390 lb and a gross of 1,560 lb -- which is very easily exceeded with just some basic kitchen gear, a propane tank, and a battery, let alone any water or camping gear. All of that is on a Dexter axle rated at 1,800 lb.
As packed away for the winter, it probably weighs every bit of 1,500 lb. It also lives outside. Add 12-18" of wet, heavy snow on top of the trailer cover, and that's another 500+ lb. It is over the axle rating with that snow load, even if in 'dry weight' form.
I may have just convinced myself to go put jack stands under that axle. I hadn't actually calculated the snow load before and seen just how bad it is.