Overall, yes the current literature supports the statement that a high BMI increases the severity of COVID-19 but it is not without opposing data, even in some of the supporting literature.
In Kim, et al. (2021) they state that
the obese had a higher chance of requiring mechanical ventilation and were statistically more likely to die than those without obesity (or being underweight).
In Soeroto, et al. (2020) they state that
Adult COVID-19 patients with poor outcome had higher body mass index (BMI) with mean difference of 1.12 kg/m2 (P < 0.001) and Obesity (BMI ≥30 kg/m2) was associated with composite poor outcome with OR = 1.78 (P < 0.001).
But counter research exists.
For example. The title of this paper tells you all you need to know but this study is only a small group of critically ill patients with COVID-19.
Kooistra, et al. (2021)
A higher BMI is not associated with a different immune response and disease course in critically ill COVID-19 patients.
I prefer in these situations to examine meta-analyses and Malik, et al. (2020) state that,
the results of the current study show that BMI plays a significant role in COVID-19 severity in all age groups, especially the older individuals.
Kim, T. S., Roslin, M., Wang, J. J., Kane, J., Hirsch, J. S., Kim, E. J., & Northwell Health COVID‐19 Research Consortium. (2021). BMI as a Risk Factor for Clinical Outcomes in Patients Hospitalized with COVID‐19 in New York. Obesity, 29(2), 279-284. https://doi.org/10.1002/oby.23076
Kooistra, E. J., de Nooijer, A. H., Claassen, W. J., Grondman, I., Janssen, N. A., Netea, M. G., ... & Pickkers, P. (2021). A higher BMI is not associated with a different immune response and disease course in critically ill COVID-19 patients. International Journal of Obesity, 45(3), 687-694. https://doi.org/10.1038/s41366-021-00747-z
Malik, V. S., Ravindra, K., Attri, S. V., Bhadada, S. K., & Singh, M. (2020). Higher body mass index is an important risk factor in COVID-19 patients: a systematic review and meta-analysis. Environmental Science and Pollution Research, 27(33), 42115-42123. https://doi.org/10.1007/s11356-020-10132-4
Soeroto, A. Y., Soetedjo, N. N., Purwiga, A., Santoso, P., Kulsum, I. D., Suryadinata, H., & Ferdian, F. (2020). Effect of increased BMI and obesity on the outcome of COVID-19 adult patients: A systematic review and meta-analysis. Diabetes & Metabolic Syndrome: Clinical Research & Reviews, 14(6), 1897-1904. https://doi.org/10.1016/j.dsx.2020.09.029