A burning pain upon penetration is a classic symptom of vulvodynia (also known as vestibulodynia). The cause is not well understood, but it's more common in low-estrogen states--like menopause. The diagnosis is made by a careful examination of the area near the opening of the vagina or hymen; the area appears somewhat reddened and even touching lightly with a Q-Tip will cause discomfort.
I have seen significant improvement with "re-estrogenizing" the vagina, which is done with prescription localized estrogen that is absorbed only in the tissues in that area and does not circulate in significant levels throughout the body. Another successful option has been a compounded (custom-formulated by a pharmacist) topical combination of estrogen and testosterone, applied to the area twice a day for 12 weeks. These two options can also be beneficial in combination.
Finding the right practitioner who is familiar with this condition is critical. A gynecologist will be most helpful, and I recommend finding one who is NAMS (North American Menopause Society) certified and in your area by searching their website by zip code.
It is highly likely that this can be successfully treated and sex will be comfortable again!