In the Name of God, the Compassionate, the Merciful.
Hi!
A: What’s going on?
B: Nothing really. How about you?
A: A lot, like the party I’m having on Friday.
B: Well, that’s cool.
A: Will you be able to make it?
B: I’m busy this Friday. I’m sorry.
A: What do you have to do? ←
B: I’m having dinner with my family
A: Maybe you can come next time.
Is “have to” a modal verb here?
If yes, what does it mean?
Thank you
“What do you need to do?”
“What do you have planned?”
Thank you so much, Arinker 
Very nice.
First of all, the sentence 'What do you have to do then?" is completely out of place in this dialogue. It is awkward, moreover, it sounds a bit like Dutch-influenced English; “Wat heb je dan te doen?” of “Wat moet je dan doen?”
Allow me to say that the entire dialogue is a mess.
As far as ‘have to’ is concerned, yes, that’s a modal auxiliary verb or a modal aux, expressing obligation, necessity and requirement.
Regarding the dialogue and your answer, they feel disjointed and sound a bit adolescent, if not juvenile, whichever, in my opinion, this forum used to be extremely lively and sparkling with new ideas, questions from people who had to deal with real grammar issues, but now it looks like it has been colonised by Rezaforu, Arinker, NearlyNapping and his alter egos like ‘Bad Luck’, that’s one of them, if I’m not mistaken.
Last but not least, this platform is meant to communicate with people…to share your knowledge of English with others who are interested in the lingo. It seems to me that you’re using it as a means of doing foul and dirty business. I don’t think you and Rezaforu should be allowed on this forum. The only thing you do, as I’ve told you before, is creating chaos and you seem to be enjoying it. But be that as it may, nothing lasts forever…