How do you store cooked pasta without losing the bite?
The true aficionado would claim that you actually don't store cooked pasta, but for those cases where this isn't viable, you have a few options:


Pasta that will be cooled and re-heated later, e.g. left overs
Drain well, cool, refrigerate. The pasta will stick, but you can dump it in hot salted water foe a bit to re-heat where it will unstick again. Not perfectly al dente, but probably ok. (Slightly undercooking pasta can mitigate this a bit.)
Pasta that should stay warm, e.g. on a buffet
Add a bit of fat (olive oil, butter...) to coat your pasta. Keeps it from sticking. Critics claim that it also keeps the sauce from sticking to the pasta. Adding the sauce can have a similar effect, but you risk soggy pasta.
Pasta that should stay cold, e.g. pasta salad
The only case where rinsing pasta is kind of allowed. Side effect: cool pasta (well drained after rinsing) can be mixed even with heat-sensitive dressings right away, avoiding stickiness issues again.