Does low sex drive make sex painful?

Lower sex drive does not directly make sex painful, but you are asking about two different symptoms that are related to each other.  Although sex drive can be decreased for many different reasons, some physical, some psychological, and some related to your relationship, low sex drive and painful sex for women are an indication that either her estrogen or her testosterone may be too low.  This happens naturally with age, especially after menopause. But it can also happen with use of hormonal birth control methods.  Since testosterone is responsible for both sexual desire as well as maintaining the woman's erectile tissue and allowing her to lubricate properly with sexual arousal, decreased levels of testosterone can severely impact a woman's drive and feeling during sex. If a woman is not properly lubricated during sex, either naturally or by using a lubricant, sex could be painful for her.
There are women who have pain related to impairments in their estrogen or testosterone levels, or for other reasons such as vaginismus (contraction of the muscles at the opening of the vagina), vestibulodynia, or certain infections.  Once a woman starts to experience pain on a regular basis, she begins to avoid sexual activity because she wants to avoid the pain.  Consequently, her desire will begin to drop as well.