Any well made igloo or snow cave should have temperatures somewhere just at or slightly above freezing. If you run a stove the excess heat will melt water, which then either drips on your face, runs down the walls or re-freezes.

Avoiding "drip points" is critical in both snow caves and igloos. The curved roof should not have any "stalagmites" where melt water will collect and drip. Any free water should run freely along the concave roof and down the walls.

Also, it takes quite some discipline to avoid powdery snow to invade just about everything inside. Many experienced winter campers will carry a small brush to remove snow from clothing and equipment. But then again, many experienced campers won't bother - they either don't see this as a problem or just ignore it. Snow doesn't stick as well to goretex clothing either - maybe that has something to do with it.

A tarp will provide a much larger "snow free" area than your sleeping pad and makes it much easier to keep the equipment, clothing and sleeping bag snow free. I always bring my bivy bag which unzips to an excellent ground sheet anyway, so I'm well covered.

Another aspect is that if it's cold, "free" water will freeze when the heat source (you) is removed from the igloo/cave. Which means there will be an ice panzer on the inside. Take precautions to ensure enough ventilation if you plan to sleep in the igloo/cave another night.

Writing this, I realize it is time I go snow caving again - my winter camping has mainly been with a big tent and a wood stove.

And I really recommend everyone with the opportunity to build an igloo and a snow cave. It's great fun - but building an igloo is rarely a viable option for building an emergency shelter. Except in some limited circumstances. A proper snow cave (with sleeping benches and so on) also takes long time to dig, but a "cave" can be very much simplified. Combined with a bivy bag it doesn't take long to dig something that could save your bacon.