Informally, you can simply say:

Allez-y

Vas-y (if you are on a "tu" basis with this person)

More politely:

Je vous en prie / je t’en prie

More formal, if the person wants you to go first and you want to insist:

Je n’en ferai rien

(Note: I have some doubts about using s’il vous plait in such a situation. I come from a region (Belgium) where we use it a lot more than in standard French, and yet I would personally find it a bit awkward in that particular situation. Some may disagree.)

You could simply go with

Je vous en prie / Je t’en prie

Which is quite formal and unisex. Keep the second one for people you say "tu" (=tutoyer) but still want to keep some kind of formality. I could use it with my boss for example, which I call by his firstname and say "tu", but still keep some kind of distance/respect 🙂

Other formulas :

Après vous/toi ! (neutral, meaning "Je passerai après vous")

Honneur aux dames ! (To use only with women, or with male friends if you wanna tease them a bit 🙂

Honneur aux anciens ! (To use with elders, with caution. I use it almost exclusively to tease people a bit, like a joke on how old they are. Most people you don’t know that well could be offended though as you’re basically implying they’re old.

j’utiliserai :

Après vous/toi !