Many circumstances which involve statistical significance can be solved by simply plotting the data to see why this can be the case. With a four-way ANOVA, its difficult to really determine what is going on here, but this is the only reasonable advice one can give with the little context given in this question. An example can be found below, where I plot the ToothGrowth data in R, which has at least two factors. The result is easy to determine...the dosage of Vitamin C has a linear effect on tooth growth. That is to say that as dosage increases, so too does tooth growth. This does not seem to change by the dosage type (supplement or orange juice), as shown in the plot.
Personally, I think linear regression with factor variables such as yours is inherently better than ANOVA. Some discussion around this point can be found in the references below.
Finally, for your main question:
I would like to do the follow-up analysis for the interaction between A and B at each level of C. Do I need to add the between subject variable in the follow-up tests?
You should include post-hoc tests regardless of whether or not there is a statistically significant effect from the model (see relevant post here), and most stats programs automatically generate these based off the models that are entered.
References
Nelson, L. R., & Zaichkowsky, L. D. (1979). A case for using multiple regression instead of ANOVA in educational research. The Journal of Experimental Education, 47(4), 324–330. https://doi.org/10.1080/00220973.1979.11011701
Plonsky, L., & Oswald, F. L. (2017). Multiple regression as a flexible alternative to ANOVA in L2 research. Studies in Second Language Acquisition, 39(3), 579–592. https://doi.org/10.1017/S0272263116000231
