Ike can get you out of his face easy, and his attacks hit like a truck. If we're talking customs, Ike's side-b 2 goes very fast and doesn't stop upon hitting a player, and he has a paralyzing counter that on low percents can give him a free grab, and on higher percents potential usmash for the KO. His ftilt and utilt are kill moves, and his dtilt launches you in the air which is pretty unfavorable for Falco. His jab kicks you out of his face while dealing hefty damage (though our jab is faster), and if we're recovering, Eruption can get us if Ike knows the timing (it can even kill if he was charging it.) If Ike is using Tempest, he'll try to push us away from the ledge with its windboxes and since it gives Ike a little bit of height he can go off-stage with it and still get back.
When we get Ike off-stage, his recovery will depend on where he is with respect to the ledge. He'll use Aether from below (we can pester him with Reflector if he goes up, trust me the Ike can't do anything against this), and if he uses Aether Drive it lets him travel diagonally so be wary of the sword hitting you in the middle of the stage. From above, he'll want to use side-b to get back on, either sweetspotting the ledge or letting momentum propel him on stage. I don't know for sure if fsmash can hit him out of it if he aims for the ledge, or if we can read it with dair. If he goes for on-stage he'll be stuck in freefall until he lands, so if we can reach him (say on FD) it'll be a free grab or even jump-cancel usmash if we can hit that. Another option is to interrupt his side-b by getting hit by it off-stage. It's a good idea at low percents because if he's far enough he can't recover even with follow-up Aether. Be wary though that if you try to do it at high percents it can kill, and this won't work if he's using Side-B 2 or, if he's close enough to the ledge, Aether Drive (diagonal Aether.) Side-B 3 I haven't tested for recovery but it's slow so it doesn't get much use.
Ike has a counter, and it's one of the stronger counters knockback wise. Default counter hits hard, I already covered paralyzing counter, and his third counter has a bit more startup but kills much earlier than default. If you want to punish him, mix up your timings to catch him off guard if he counters a lot, and especially keep your grab game on point.
Ike gets a lot off a grab, dthrow sends you up which he'll follow with fair or uair depending on your DI, and Aether at low percents. Fthrow and bthrow are used (by me anyway) to send characters off-stage so they'll see more use near the ledge, and uthrow isn't that useful since dthrow does it better IMO, but Ike might use it to avoid staling if he's been using dthrow a lot. Dthrow can kill around 150% fresh, so be careful.
Ike can't effectively punish lasers from a distance so use them to rack up damage from afar. I highly recommend you pick stages like FD, Smashville and Duck Hunt against Ike. Less platforms give him less maneuverability, and the length of those stages let you stay grounded which is highly beneficial.
You pop Ike in the air, he doesn't have many option. If he's defensive and goes for air dodge or a counter, bait it and retaliate with uair or your aerial of choice that can hit. If he's offensive he'll probably come swinging down with nair because of how its hitbox circles around him, or fair due to it reaching the bottom near the end of the animation. More likely than not though, Ike will play defensive. Take into consideration his DI and playstyle when coming up with a way to manipulate his actions, and keep him airborne as much as possible.
Don't be reckless, as Ike can tack on ~25% with a grab and an aerial, and finish him off as quick as you can. The longer you let Ike live, the more he'll enjoy the benefits of rage. Rage gives his kill moves a boost, and can flip a match around if you're both at high percents. Falco can't afford to get punished by Ike, so play patient because Ike can't be reckless either, since most of his kill options have enough endlag to get punished by spotdodge to grab or something similar.
I welcome anyone to discuss what I've said, as this is what I have to offer from playing both characters for a great period of time. I'm not the best Ike player by any means, but I know my way around him.
Edit: Upon some reflection, I've decided that laser and default Reflector are our best tools for edge-guarding Ike. If he attempts to side-b from a distance, fire lasers to interrupt his charge. Depending on how far away he is, this most assuredly shuts him down right there. If he's using Aether, take his current location into account. With default Aether, his best location is as far below the ledge as possible to grab onto it. You'll want to interrupt with Reflector as much as you can, it racks up damage for free and Ike can't do anything about it until you let him grab the ledge again. Don't get too close either; when he tosses his sword up the hitbox will push you too far away to hit him, so find the right distance. If he's using Aether Drive and recovering from below (think hugging the stage), he has to do it at the optimal angle. Too low and he'll miss; too close and on some stages he'll get caught under the lip or won't gain enough height. Too far and he'll miss, obviously. Stay on guard when an Ike is using Aether Drive, as the diagonal angle the sword is tossed means it can hit you a distance from the ledge.
Also, Ike has super armor in his recoveries; it'll take a strong attack to knock him out if you want the stage spike. Don't go for the bair unless you know the knockback is enough that it would kill normally, otherwise you'll just damage him and put yourself in a difficult position. Also, Ike can't use Aether to camp the ledge; after 6 uses without getting back on stage the Aether will simply fail to grab the ledge. Finally, take note of whether he tries to for a ledge regrab with Aether; that's your chance to run in, shield Aether and strike with anything from dsmash which launches him to where he can't recover, or maybe a low-angled ftilt or even uncharged fsmash if you've got the time.
Finally, Aether spikes coming down so be careful; if Ike hits you with his side-B in the air he can immediately follow-up with his up-B.