To my knowledge, no formal work has been done on the economics of EIP1559 under eth2. Structurally, the overall schema would likely be identical. However, PoS would certainly change the weight of some of the parameters.
If eth2 features 1024 shards, overall gas consumption would increase considerably. At the same time, increased throughput would likely make network fees much lower. Combined, those two factors could potentially balance things out and ultimately lead to a proportionate amount of gas being burned. In theory, PoS could also solve the aforementioned issue of off-chain agreements, but only if validator cartelization does not occur.