Although it is never specifically said, the implication is very clear that Tom and Daisy left Chicago because of one of Tom's indiscretions.
The chain of evidence for this fact begins in Chapter four when Jordan tells Nick of Daisy's history:
A week after I left Santa Barbara Tom ran into a wagon on the Ventura road...The girl who was with him got into the papers too because her arm was broken - she was one of the chambermaids in the Santa Barbara hotel.
This quote proves that Tom was dallying with a maid only weeks after returning from his honeymoon with Daisy. Jordan goes on to share that eventually Tom and Daisy "came back to Chicago to settle down...They moved with a fast crowd, all of them young and rich and wild..."
Later, in Chapter seven, when Gatsby confronts Tom in the hotel scene, Tom and Daisy again make reference to their sordid past. Tom says to Gatsby "Once in a while I go off on a spree and make a fool of myself, but I always come back, and in my heart I love her all the time." To this Daisy replies hotly, "You're revolting" and says to Nick, "Do you know why we left Chicago? I'm surprised they didn't treat you to the story of that little spree."
So, although the reason for leaving Chicago is never expressly stated, it is readily apparent throughout the text that Tom and Daisy had to leave Chicago because of one of Tom's many indiscretions.