using a basal thermometer while on birth control?

Answer 1

That's not going to work too well since hormonal birth control supresses ovulation.

So if your ovulation indicators (like temperature) are off because of birth control, there is not much of a point in taking your temperature.

The only thing it would be useful for is if you got pregnant while on the pill. Your basal temperature would stay elevated throughout pregnancy so your thermometer would be useful as a sign of conception.