It seems unlikely that the phrase, in practice, will be going anywhere. When President Hassan Rouhani returned from New York recently, a few dozen hardliners showed up and egged him. There's unlikely to be much political will for taking on those factions of political society.
The slogan still forms a pillar of the Islamic Republic's revolutionary values. Even moderates and some reformists are still indirectly referring to the U.S. as the enemy, so that should be a strong indication of where those values lie. Not to mention the manpower it would take to scrub clean the many billboards and walls in Iran that have some version of "death to America" written on them. What's more likely is that it fades away gradually and becomes a less prominent component of organized events.
There was a time when "death to Russia" was also a chant, but it went away as Iran's relationship with Moscow improved. Here are some other chants that have rung out at times in the Islamic Republic's 34-year history: "Death to England," "Death to France" (briefly), "Death to Israel," "Death to Saddam" (not Iraq) and "Death to the Mujjehedin Khalq" or MEK.
It's also important to remember that most official translations have it not as "death to America" (mar bar Amrika), which is what people say, but as "down with America," perhaps so as to not cause so much offense. It's a subtle difference but a notable one.
These institutionalized reminders of the Islamic Republic's early days are probably not going to be cut outright anytime soon, but they don't have the weight that they once did. It can be tempting for outside observers to compare what's happening in Iran now to what happened in the Soviet Union in 1989. But there's no indication that there will be any dismantling of the political-ideological system in Iran or even significant changes to it, as happened in the Soviet Union, so there won't be those same sorts of climactic moments.
Still, this discussion about the chant is foreshadowing things that could be to come. In the end, it's most likely that this will play out between the Iranian and American leaders, with lots more of these mini-moments like the phone call between Rouhani and President Obama. But there will still be chanting.