The exhaust from a typical ICE wouldn’t have enough pressure to inflate a tire, so you’d need a compressor. Of course if you had a compressor you’d just use clean air.
If for some reason you used a compressor to compress exhaust gases to fill a tire, it would mostly be the same as filling with air at first.
Exhaust gas is mostly a mix of carbon dioxide and and water vapour, with small amounts of oil residue, and other organic compounds. The water vapour will condense as it cools likely leaving some liquid water in the tire, which won’t cause immediate issues but will cause vibrations which will accelerate wear not just on the tire but possibly the entire suspension.
The organic compounds will cause the rubber to break down over time and the tire will wear out sooner.