In theory, yes increased turnover/expansion of immune cells in inflammatory contexts can lead to increased frequency of oncogenic mutations that turn into hematological cancers. Each round of cell division has the possibility to introduce oncogenic mutations (mutations in genes that transform a normal cell into a cancerous one) as a result of DNA replication errors. But there are also many mechanisms to correct such errors or kill off cells with oncogenic mutations. So the chance of cancers arising solely from this mechanism are low and there are a number of other factors that play more substantial roles in cancer formation.

The conditions you listed are associated with chronic inflammation. Chronic inflammation has been widely established as a risk factor for many different cancers. While increased cell turnover (e.g. immune cells in inflammatory settings) can potentially enhance the risk of oncogenic mutations, the inflammatory microenvironment (i.e. both cellular composition and factors secreted by theses cells in inflamed tissue) itself is likely a much more substantial driver of cancer formation. For example, remodeling of the extracellular matrix (ie the scaffold that forms tissue structure) is an important feature of inflammation but is also critical for metastasis and/or formation of blood vessels (angiogenesis) that could feed a growing tumor. Immune cells in inflamed tissue can secrete MMPs (which break down and remodel the extracellular matrix) and angiogenic factors. While the immune system/inflammation is essential to kill cancerous cells in tumors, some subsets of immune cells present in the tumor microenvironment (TME) secrete factors that suppress the anti-cancer immune response. Chronic inflammation can enhance or induce these suppressor subsets. Here's a nice review on the mechanisms by which inflammation drives cancer formation.

For your question specifically, there is a JAMA paper that reports an association between atopic eczema and lymphoma risk. It's not a particularly strong association, but they did find that the lymphoma risk correlates with eczema severity. Given that the potency of the inflammatory response determines severity, this is in line with what I mentioned above. In theory, controlling inflammation (which is the principle behind most eczema therapies) should reduce the cancer risk. But broad immune suppression increases the risk of infection. Infection with Epstein-Barr Virus (EBV-one of the causative agents of mononucleosis) is a well-recognized risk factor of development of both Hodgkin's and Non-Hodgkin's lymphoma. EBV infects lymphocytes and encodes proteins that contribute to cancer cell transformation and augment the TME to make it more suppressive. Other viruses have also been associated with this before. EBV driven lymphomas are a potential complication in transplant patients (which receive potent immunosuppressant drugs to increase transplant success).

So overall, yes chronic inflammatory diseases like eczema or asthma can potentially promote cancer but not (solely) through the mechanism you describe. I hope I've been able to help your prospective study (which I think you should definitely write, its a really neat hypothesis)!