Charlemagne was probably born between 742 and 747, as he was old enough to be present and speak at an official reception for the Pope in January 754; Irene was born around 750-2 as she was married in 768 (to future Emperor Leo IV) and had a son, Constantine VI, in 770. She had no other children. From this I would assume that they would not have children; but I cannot see the marriage working in practical terms.
The two regimes, West and East, were not yet at religious odds as the dispute between Catholics and Orthodox over theology - specifically the Western official creed adding on the new words that the Holy Spirit proceeded 'from the Father and the Son' to the original wording 'from the Father' - did not start up until the 860s and the definitive breach between them was only in 1054. But there was intense elite Byzantine hostility to the Westerners as 'barbarians' and anger at the Frankish assumption of the title of 'Emperor' in 800, as 'the' Emperor was supposed to be the 'Roman' ruler , ie of the Roman Empire (now just the empire in the East as lineal descendant of the early Roman emperors) not the ruler of former Roman lands in Italy and Gaul/ France. Charlemagne was thus a 'usurper', and was not recognised as Emperor by Byz until 812; and the two states were clashing over control of southern Italy, reconquered by the East from the Goths in the 530s and since then ruled from Constantinople - the local Lombard states around Benevento and Spoleto, former Byz vassals, were now under demands from Charlemagne to do homage to and accept rulers set up by him, as the new lord of Rome and the conquered Lombard kingdom in Pavia/ Lombardy, and Benevento elite refugees from his rule had fled to Constantinople. Both sides also claimed overlordship of the lagoon state of Venice, a surviving part of Byz Northern Italy (lost by them to the Lombards in 751) but claimed as a vassal by the Franks - when Venice resisted this later in the 800s Charlemagne sent his son Pepin, his viceroy of Italy, to besiege it.
Even if goodwill and a determined effort by both sides can sort this out, possibly due to mutual need to co-operate and sort out the Avar nomads in modern Hungary as dangerous raiders (and then the Bulgars too?), I cannot see Irene daring to leave Constantinople - even by sea, for a few months only, for a formal marriage at a mutually agreed place in S Italy or Sicily, though Charlemagne could get there easier by land and trust his eldest son and main heir Charles to keep Francia secure. Irene was not too secure in her capital, being an outsider (from Athens)and having overthrown and blinded her incompetent and erratic son Constantine VI in 797 - his family had held the throne since 717 and were usually popular with the powerful army as the champions of Iconoclasm, unlike Irene who had restored the worship of icons in a controversial move in 786-7 as regent. C VI had thrown away his popularity with clumsy campaigns and the army had deserted him in 797, but Irene had no heir or secure male political/ military elte allies and had to rely on a small clique of competent but mutually squabbling ministers, mostly palace bureaucrats and court eunuchs - all eyeing up the succession for themselves or their relatives. She ended up overthrown by her treasurer, Nicephorus, in 802 and packed off to a nunnery.
So if she leaves her capital and then gives too much away to the Franks she would be targeted by her elite and probably fall in a coup; probably a proxy, official marriage , in Constantinople to a Frankish envoy standing in for Charlemagne, and a diplomatic alliance based on this is the best that could be hoped for. It might last if she did not give too much away (eg in southern Italy) and Pope Leo helps her on that, and if she is more skilful or lucky at home - eg if the rampant Bulgar threat to the Byz lands in southern Thrace can be dealt with by the Franks sending in a large army SE from the middle Danube to take the Bulgars in the rear and Irene can then reoccupy parts of Macedonia/ the Hebrus valley and force the Bulgars there to turn Christian and supply her with troops. Then she could cut a deal with one of her squabbling advisers to be her heir if he supports her long-term, possibly Nicephorus by marrying her young kinswoman Theophano off to his son Stauracius and promising S and T the succession. If the Bulgars are defeated c. 802-5, then they do not kill Nicephorus and mortally wound Stauracius in 811 as in OTL; the two of them could last on the throne through the 810s and 820s. But this is the most that any realistic alliance could get out of the marriage.