If the bird is a toon then there's no need to worry about a kiss going wrong with them as their beaks will warp and stretch to make the kiss the work:
post #2791125

But if they're semi-realistic, yet still functions like a toon bird's beak, as in they're still flexible and even have teeth, and you don't want to do just tongue only, then to kiss a bird character you need to decide on which part of the beak you want to wrap your lips around on and go from there.
While it is a childish way to describe it, but I think "A kiss is like a hug for lips" is more true for bird characters than any other type of anthro and the couple should focus on the feelings(sense of touch) their lips/beak feels against each other. I'd like to imagine that depending on the beak, the beak feels really smooth in general but hook like ones would feel interesting to explore with your mouth.

However if the beak is realistic, I'd have to say give up on a normal style of kissing since the most beaks function more like scissors and things to cut and tear into meat and other things to eat.

But depending on the setting, I think the bird characters should do something like a Paper Test where they poke the tip of their beaks through some kind of normal office/school paper and see how much strength they need to bring before it tears through the paper.
If the paper tears through it easily then it's just not possible for a bird character to kiss anyone without hurting them, but if it takes a bit of strength to make a tear happen then that means the bird character has a chance of making a kiss happen and just need to remind themselves in their partners about the dangers and work they need to do when they kiss each other.

And those are my ideas on how kissing bird characters work when they don't have functional cartoon lips on the tips of their beaks.