She did ‘Umrah and forgot to cut her hair, and her husband had intercourse with her
Question

I was with my wife in Mecca to perform ‘Umrah. After we returned to our home we had intercourse. Then my wife remembered that she did not exit Ihram. What is the ruling on this?.
Praise be to Allah.Shaving the head or cutting the hair is one of the obligatory duties of ‘Umrah, and the one who forgets it should do it when he remembers. If he did any of the things that are forbidden when in ihram before that, out of ignorance or because he forgot, then he does not have to do anything, according to the most correct scholarly opinion. 
Based on that, your wife has to cut her hair now, and thus she will exit ihram from her ‘Umrah, and she does not have to do anything because of the intercourse that took place, because she only did that because she thought that she had exited ihram following ‘Umrah. 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said concerning a woman who did not complete her ‘Umrah:
With regard to the things she did that are haraam when in ihram, let us assume that her husband had intercourse with her at this time, and intercourse when in ihram is the most serious of the forbidden things. But she does not have to offer any fidyah because she was unaware (of the ruling). Every person who does one of the things that are forbidden in ihram out of ignorance or because he forgot or was forced to do it, does not have to do anything (i.e., offer any fidyah). 
From Majmoo’ Fatawa Ibn ‘Uthaymeen, 21/351. 
Some scholars are of the view that having intercourse after doing saa’i and before shaving the head or cutting the hair means that the fidyah is required, even if one forgot or was unaware of the ruling, but it is fidyah in the sense of being optional, like the fidyah offered in the event of an ailment in the scalp, i.e., the fidyah for one who shaves his head because of an ailment such as lice and the like, as is indicated by the words of Allah (interpretation of the meaning): 
“And whosoever of you is ill or has an ailment in his scalp (necessitating shaving), he must pay a Fidyah (ransom) of either observing Sawm (fasts) (three days) or giving Sadaqah (charity — feeding six poor persons) or offering sacrifice (one sheep)” [al-Baqarah 2:196]
So he should fast for three days, or feed six poor persons by giving each one of them half a saa’ of wheat or other staple food, or slaughter a sheep to be distributed among the poor of the Haram (in Makkah). 
See: Sharh Muntaha al-Iraadaat (1/556). 
Your wife may follow this view so as to be on the safe side, so she may fast for three days or feed six poor persons or slaughter a sheep to be given to the poor of the Haram. 
And Allah knows best.