Vcar & BC don't interact. The 2 issues with being on a cycle & using birth control that sort of contradict each other:
1) Often women look to var as as low-sides AAS which doesn't cause water retention. If anything, they expect to lean out lots on it. Var isn't a fat reducer per se. Birth control often (depending on the person, tho) regulates estrogen levels and in some cases, may also promote (or maintain) estrogen-related fat depositing & a certain amount of water retention. So while combining var & BC, there is no interaction between the two that negates the intended function of one or the other, but the results as far as trying to achieve significant fat reduction may be compromised. Again, this depends on the person and is also a bigger deal if you are pursuing the var cycle with the intended purpose of preparing for a competition where you are seriously focused on reducing fat & water retention while building & maintaining good quality muscle.
2) Birth control is often used to help establish regular periods, cramping, etc. so the period is regular & predictable. AAS sometimes makes it appear that a period has stopped. So then you start to wonder if your BC is actually working. Another item related not to the AAS but to the dieting often associated with the ultimate purpose of cycling to build muscle & lean out is that at a certain point in body fat reduction (e.g. 11% bf & lower), the period may stop.