Its an interesting question that you raise. I guess a lot depends on how much the normality assumption is being violated, many people would just quote that the test is robust and leave it at that. However, I think you are right to question it. Maybe this paper will help you:
Lix L. M., Keselman J. C., Keselman H. J. (1996). Consequences of assumption violations revisited: a quantitative review of alternatives to the one-way analysis of variance F test. Rev. Educ. Res. 66, 579–620. doi: 10.3102/00346543066004579.