As mentioned, there are 2 issues with running a snowmobile on bare ice with no snow.
#1: Engine overheating. Liquid cooled snowmobiles need snow to cool them, this can also be an issue on hardpack trails. This is a non-issue with fan cooled snowmobiles. You would need a fan cooled sled, no questions asked.
#2: Slides not being lubricated and wearing fast, possibly even melting to the track in extreme cases. Ice scratchers help somewhat and another option is squirting dish soap on them every couple miles.
I use my snowmobile in no snow conditions all the time early and late ice. I need it to fish where I want for most of the season and don’t have a four wheeler. It’s fan cooled so overheating is no worry and I use the dish soap to help protect my slides. I only make short runs (less than 2 miles), watch the slides for wear, and change them when needed. A new set is less than $30. Be careful driving and keep the speed down on glare ice, don’t touch the brake unless absolutely necessary. It’s really easy to get the sled spinning cirles and if you catch something at the wrong time, you will roll it.
If there are times in your season when a snowmobile is absolutely necessary and you don’t want to buy an atv as well then follow these steps and you’ll be fine.
If you don’t get enough snow to NEED a snowmobile then get an atv.