I'm not aware of any constant foreshadowing in Futurama. It's been pretty clear from the series that Earth is always in danger one way or another (out of dramatic necessity). Earthicans have been close to annihilation dozens of times from a variety of threats, ranging from V-GINY to the Brain Spawn to a giant ball of trash to the common cold to alien cats to global warming to the Dark One to the Beast with a Billion Backs...
Morbo's threats are clearly just a running gag not meant to indicate a specific event. The writers could eventually add an episode where they attack and are driven off, but they're probably more likely to just continue the gag as is.
If anything, the Futurama writers seem to espouse a cyclical view of history. Whenever deep time travel is shown (e.g. when Fry gets frozen the first time or when they ride Professor Farnsworth's time machine), there is usually a time-lapsed shot of human civilization being destroyed and rebuilt multiple times in quick succession. Sometimes it's alien conquerers, sometimes it's a comet or other natural disasters, or it could be nuclear war. Either way, even if humanity is reduced to a dark-age-level culture, it is gradually rebuilt back into an advanced futuristic society. Even when the Sun burns out and the universe goes cold, everything is eventually reset with a new big bang and the whole history of the universe repeats itself (the time travel episode has them going around this cycle not once, but twice).
If you wanted to, you could pick any of the future cataclysmic events as the one being "foreshadowed", but it would be purely arbitrary. In Futurama, humans are constantly being endangered and constantly surviving or recovering from such events. The Earth has had its entire atmosphere burned up during one Christmas special. The same episode also had a different Robanukah skit showing humanity long extinct from an unspecified cause. And the whole universe was also destroyed by a tear in space-time in one episode. A Valentine's Day episode showed life on Earth wiped out by an exploded diamond meteor.
Clearly no specific eschatology is planned for the Futurama universe. Species are simply destined to become extinct and replaced by others (e.g. the evolutionary descendants of humans in the time machine episode), and no planet exists or supports life forever. What finally does the protagonists' civilization in is ultimately insignificant. It's just another iteration in the cycle of societal death and rebirth depicted in the series.