Are there any aircraft that have a limit on the amount of ice or snow that can be on the aircraft when stationary and on the ground?
I'm not aware of any limits every being published on transport airplanes. On large aircraft, it's not too big a deal structurally because the bending loads imposed by a snow cover are not significant, since bending load applied by a couple of feet of snow (which weighs about 40lb/sqft) on such a large structure, designed to support 100+ lbs/sqft, is relatively mild.
On little airplanes, this can be a problem however. Let 2 feet of snow, or 40 lbs/sqft, accumulate on a light aircraft or ultralight with large wings, designed to support maybe 5-15 lbs/sqft of negative load, and you will likely find the wing outboard of the strut is permanently bent down or has collapsed when you want to go flying. Nature's just done a static load test on your wings and got carried away.
On a metal airplane, you at least take comfort in the fact that if the wing is not permanently bent down after you remove the snow, you're good to go.
On an airplane with wood spars, this is more of a gamble, because wood structures that are overloaded beyond the compression limit of the side in compression (the spar below the strut attachment bolt in the case of snow load) without overloading beyond the tension limit of the side in tension (above the strut attachment bolt, where it just breaks), can suffer from a dormant compression failure in the wood itself.
Compression failures are where the cellular structure of the wood fibres partially collapses. This can leave the wood apparently undamaged (unless you can inspect the area to detect the tell-tale fine line across the grain), but weaken the wood in tension. So when you go flying, and the wood below the spar attachment is loaded in tension again and has lost some fraction of its ultimate tensile strength, and you hit a bump that applies some extra G, whoops!
Anyway, if you own a light aircraft with wood spars you need to be aware of this and be extra vigilant about preventing large snow loads on the wings. I'm not really keen on wood spars on light planes for this reason. It's a reason you never buy a used Classic with wood spars that's had a wing tip repair (from the tip hitting the ground during a groundloop). A friend bought an Aeronca Champ once and found his right wing had a wood compression failure above the spar bolt, caused by a wing tip strike, when he took it apart to recover it, after flying it about a year. He was lucky; any significant negative G in that condition would have been fatal.
You could work out a limit for yourself if you know the wing loading and structural G limit for your own airplane. So if your wing loading at gross is 10 lb/sqft, and it's good for negative 2 Gs in flight, you can estimate that the wing can tolerate 20 lbs/sqft of snow accumulation, or about 1 foot thick and wouldn't want too much more than that to accumulate.
I've seen a Boeing 727 parked at an air museum that tipped over so that the rear fuselage was on the ground, due to the aircraft being heavily covered with ice. Presumably it would be desirable to avoid that event in an operational aircraft-- to avoid cosmetic damage to the bottom of the rear fuselage if for no other reason.
No, However to protect airplane from icing , it is recommended to park in closed hanger and the geographical area where icing occurs frequently they keep airplanes batteries always for charted state so that it is easy to take the plane for flying after normal preflight procedure.
For transport category airplanes, there are often manufacturer recommendations for storage of aircraft, and engines. The level of storage (length of time and environment) would determine what precautions are recommended. In the case of transport aircraft I don't know if those requirements are part of the approved maintenance program or not. Standards for maintenance requirements are developed at the industry level by Maintenance Steering Group -3.