Irulan is the
sole * heir of ruling House Corrino and prior to the events of "Dune" Leto Atreides was gaining popularity with Landsraad (something like a council of the noble families), which seems to be divided in support of those two families.
By marrying Irulen he gained two significant advantages:
- He legally took over from the ruling Padishah Emperor as his son-in-law
- He united (to a degree) the Landsraad preventing a war.
Of course his true love was Chani, but from political point of view she was nobody, but as Lady Jessica said (which also never married Leto, allowing him to potentially have a political marriage):
"Think on it, Chani: that princess will have the name, yet she'll live as less than a concubine — never to know a moment of tenderness from the man to whom she's bound. While we, Chani, we who carry the name of concubine — history will call us wives."
Paul never had any children with Irulan, but she was treating his and Chani children as her own.
Later on similar political marriage you will see in the "Children of Dune" where
Leto II will marry his sister Ghanima, but will never have children with her (instead she will have official consort, who will father her children).
Of course, as the OP mentions, Paul HAS the power to force his will on the Landsraad and marry whomever he'd want, but it would lead to prolonged war with those that would see him as an usurper. Such political marriages against the will of heart were quite common in human history.
@Richard's comment is actually an answer, I'll just extended it a bit.
* While Irulan is not the only heir to the throne (as pointed by @Wrzlprmft) she is ONE of heirs (and please remember that ruling families wouldn't want to have too many children trained to become a heir to avoid the possible civil war) and by marrying her Paul has full legal right to take over from her father when he'd resign. And Muad'Dib can be very persuasive with all his army outside of the palace...