Informally, you can simply say:
Vas-y (if you are on a "tu" basis with this person)
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.
Après vous/toi !