[Beanamir of Gondor]

Wow. This is such an incredibly complex question... I mean, this is actually like a Civil War historian asking "What if Lee hadn't ordered Pickett's Charge at Gettysburg?" The entire course of the war would have changed--the Confederacy might not have won instead, but there certainly still would have been disastrous consequences for the entire country.
In all likelihood, if Aragorn, Legolas, and Gimli had gone after Frodo and Sam, Merry and Pippin would have been all right with Treebeard.... but then the Triumphant Trio would have followed Frodo and Sam across the Dead Marshes rather than aiding Eomer and Theoden. Gandalf would have only been able to do so much (especially with Eomer being banished), because it was Aragorn who inspired Movie!Theoden to fight well (and with the Movie!Elves) at Helm's Deep. So yes, the Rohirrim and the people of Edoras might have died at Helm's Deep without the Incredible Surfing Elf and his companion the Comic Relief Dwarf, not to mention Strider.
Really, the possibilities and consequences are endless. And Boromir88 is right: most likely Gollum would have had no part to play in the drama, and the Ring might have ended up in Aragorn's hands, or someone else's, once in Mount Doom. That's considering if the fivesome made it past the Black Gate, or through Shelob's Lair!
Then think of the consequences for Gondor! No return of the king... no Men under the Dimholt Mountain fighting on the Pelennor... no one to save Eowyn, Faramir, and Merry from their sicknesses IF the Gondorians had managed the battle without the Rohirrim...
Augh. My head hurts. Perhaps the best idea for this thread would be to organize the possibilities right from the Breaking of the Fellowship, take it one thing at a time.
Actually, that might be quite interesting. Let's do it!