Mitt Romney's tax returns affected Romney's campaign during the Republican primary. Though there were no "bombshells" in the returns Romney eventually released during the primary, his prior refusal to release any returns created a cloud of suspicion that (temporarily) hurt his campaign. After Romney released some returns, the issue disappeared from the Republican primary. It would have remained a dead issue, but Barack Obama's campaign has been calling on Romney to release additional returns; Obama himself has released his tax returns dating back to the year 2000. For now, Romney refuses to release tax returns earlier than 2010.
How might this issue play out during the general election? An issue like this requires a "driver"--something to keep it in the news or bring it back into the news once it fades. Until Romney releases more returns, there is no "hook" for a story; there's no new information to report on. The Obama campaign can attempt to keep it in the news by repeatedly calling on Romney to release his returns, but that's all. Unless this causes a big change in public opinion, the issue will fade from the headlines, because the repeated calls for disclosure will grow stale in their repetition. For the moment, public opinion is unlikely to change very much because most voters are not yet paying attention to the race.
However, the tax return issue can harm Romney even if it is not kept in the spotlight for a long period of time. Recall the way it played out during the Republican primary: No one mentioned Romney's returns during 2011, and the subject wasn't brought up until January 2012. When the matter was raised by Romney's opponents, it had immediate impact. Romney lost South Carolina by a wide margin and polls showed Romney well behind in Florida. This loss of support wasn't reversed until Romney released some returns. Thus, it's possible for the Obama campaign to set the issue aside for now and still raise it again, closer to the election--in October, for instance.
Rival campaigns aren't the only potential "drivers" for a story like this. Once again, the Republican primary is instructive. It wasn't merely the calls from Romney's opponents that put the issue in the spotlight. Rather, the most serious heat Romney took was when the issue was raised during a Republican primary debate. The debate moderator acted as the driver. The moderator reminded Romney of the fact that Romney's own father, Michigan governor George Romney, was a loud proponent of candidates releasing many years of tax returns. The moderator then asked Romney if he would follow his father's example, and Romney gave a painfully evasive answer.
While it would be awkward for Obama himself to inject "Well Mitt, why don't you release more tax returns?" into a debate, it's easy for a debate moderator to do so. If the moderator is a particularly ardent supporter of Obama's, he might ask repeated follow-up questions to pin down exactly what Romney feels needs to be kept secret. Romney is unlikely to give an answer as weak as the one he gave when this question was raised during the primary debate, but it's difficult for him to give even a well-prepared answer without appearing evasive.
Later in the campaign season, when more people are paying attention, it will be easier for Obama to push the mystery of Romney's tax returns into the spotlight. As we saw with the "birther" controversy, something can remain dormant for awhile and suddenly, unpredictably flare up later if it's unresolved.
In summary, it will be very difficult for Romney to make it to November without releasing his tax returns. The issue can temporarily disappear or grow stale, but reappear more harmfully closer to Election Day. Even if the Obama campaign fails to get voters excited about the issue, an Obama-friendly debate moderator can bring the matter to center stage during one of the crucial debates.