I once ran an adventure where the players had to go back in time to save the biggest douche bag in the world, because the fate of the world depended on it. That was fun. And then they accidentally caused a paradox that caused clones of themselves to spill into their own timeline. Fun times.
But my best time travel adventure was a Call of Cthulhu campaign I once ran:
The story took place in the present, and was about the players traveling to Evans City to investigate a series of strange goings on at the local elementary school (the kids were having head aches, nose bleeds, and making disturbing drawings). The school was a bit of a red herring, because far worse things were afoot. Upon arriving in the city, they passed through some sort of an invisible field, and then they hit an exposed concrete tunnel with the bottom of their car, that was hidden underneath a muddy puddle of water. This tunnel was part of a hidden underground network, used by a local cult. It connected various important locations in the campaign.
But the players were now stranded in the city with a broken car, and they gradually started to experience many bizarre events. They would receive phone-calls from the future, and the caller would give the answer to a question before they asked it. They would also see phantom images of future events passing through the city at night. They could even feel and hear someone getting into their hotel room at night. They would see the door opening, the sheets being drawn back, and feel the mattress being pressed down. A time-echo of a future hotel guest, as it turned out. And every now and then there was a minor earthquake. As it turned out, the city was a few seconds ahead in time. But each day this time difference became bigger. After a few days, the city was one whole day ahead in time, and they could see future events on the news. But this time travel effect was only in effect within the boundaries of the city (this is why they passed through an invisible field on the way in). The time difference also meant that passing through the time barrier became more and more harmful to anyone attempting to enter or leave the city.
I had carefully thought out the exact rules of the time difference, specifically in regards to modern communication equipment. For example, you could easily talk to each other within the city with walky talkies, as the signal never left the city. But if you used a phone or computer, the signal would leave the city limits, and by extension pass through the time wall. This would cause time differences in the signal, allowing you to receive phone calls and tv broadcasts from the future.
Eventually this event reached a critical point, where the city became more and more displaced in time and space. Now the voices on the other side of the phone were no longer of the person you called, but terrifying screams and wails along with loud static, from another world. The city was gradually merging with another dimension, thus bringing it into our world. And there were dangerous creatures living just on the edge of our reality; invisible, yet they would become aware of the players whenever the players became aware of them. Cracks would appear in the ground as the city edged closer and closer to complete disaster, and the cracks lead the players to a local church.
There in the basement deep below the church was a strange machine, an irregular shape of unknown metals, humming and buzzing loudly. Various wires and tubes were attached to the machine, with no obvious on or off switch. The players knew that they had to put a stop to the machine, to avoid the inevitable calamity.
Now here is where this story gets really interesting:
I ran this campaign with two different groups of players. The first group tried to destroy the machine, after killing the main villain. This caused an explosion, and a time rift sucked all the players back in time, into the late 50's. They awoke in a hospital with shards of the machine embedded in their body, as those same invisible creates from another world were drawn to the fragments. This was a set up for a follow up campaign.
The second group decided to carefully unplug the machine. They avoided harm to themselves, but it was too late to reverse the effects of the machine completely. They fled the church as the town was slowly sucked into another dimension. They also managed to turn one of the main villains into an ally, as she came to understand the danger she had unleashed, and she helped them escape.