I'm not a lore hound for either setting, so take this with a grain of salt. At a guess, the gods would be a wash...so ignore them.
Eberron has wide but not tall magic. Meaning there's lots of low-level magic being used, but not much high-level magic in the setting. It's there, it's just not as common as some other settings.
Forgotten Realms has wide and tall magic. Meaning there's lots of low-level magic being used and a lot of high-level magic in the setting.
So just thinking about it from that angle, the FR are likely to push back the invasion easily at first, throwing out high-level spells that the Eberron invaders wouldn't have access to or be able to deal with effectively, but those resources would run out. Spell slots get used, material components aren't available at the corner store. There'd be a planet-wide run on a lot of rare and already expensive components.
The invading forces would be outnumbered by the inhabitants of the FR by default. You're not mobilizing the entire planet to invade another planet...but Faerûn will be able to mobilize basically anyone that can fight with an interest in defending their home. Though using history as a guide you could make a good argument for the Eberron forces turning local factions against each other to weaken various FR kingdoms, etc. To dip even more into history, there's the question of diseases. Each population would have exposure and immunity to various diseases the other has never encountered before. This would kill a lot of people on both sides.
The invaders could establish a beachhead somewhere on the planet, likely one of the Remembered Realms or the Best Ignored Realms, and move out from there. Minimizing their points of contact with the enemy and not alerting the whole planet at once. If they were to invade, that would be the way to go. Find a nice, quiet, out-of-the-way corner of Faerûn and take it over. Use the local faction's rivalries against them to make gains. Move out from there. With blitz attacks they could strike out and take over quite a bit of territory in a hurry.
ETA: Any kind of sci-fi thinking would end things in a hurry. Mass drivers. Snaring asteroids, comets, and other random space debris and simply hurling it towards the surface of the planet. If you don't care about the planet itself, the geography, or what's left to control you could be done conquering the planet by wiping out the whole population in an afternoon.