I really like the idea of roles. It helps create more interesting, balanced classes and helps convey the core ideas of a class to a player in an efficient manner. There are a few really big problems with how they were implemented in 4E that need to be fixed for 5E, though.
1) Roles need to be presented in a different manner.
One of the big problems with 4E roles is that they are presented as "each role needs to be represented to make a balanced party." this probably isn't the best way to go about it. Roles are really useful for class design because they give classes focus, and help make sure that a class has clear, intelligible strengths and weakness. As Felon stated quite well above, they prevented the situation from 3E where some classes (like the Monk) were poorly thought out and failed to contribute to the team while other classes (like the Cleric) had no weaknesses can could replace almost any other class. However, just because the class has a clear role doesn't and shouldn't mean that role is essential to a team or that every class in the same role should play the same way. Basically, roles should help players make decisions, not make decisions for them.
2) There should be more variety within any given role.
4E really had a problem with roles that the four roles were basically Fighter, Rogue, Cleric, and Wizard, and every class in a role tended to resemble those four classes, even when it was unnecessary. You don't strictly need heavy armor and a melee weapon to serve an important defensive role in a team, so not all Defenders need to be heavily armored melee combatants. Different approaches to the same solution add a lot of fun to the game.
3) We need a new set of roles. Badly.
As I just mentioned, the 4E roles were rather transparently based on the four traditional D&D classes. This ended up being a terrible set of roles. Defender and Leader work quite well, but the other two really don't. Strikers don't serve the game well because every 4E class was supposed to be dealing regular damage, so a dedicated damage class doesn't have the clear role it should have. The Controller is in an even worse spot, since it is not a very well-articulated concept in the first place (other than being vaguely wizard-like).
One thing that always struck me is that, for all the problems with the four class roles, 4E always had a great role system with its monster roles. The set of Brute, Soldier, Artillery, Skirmisher, Lurker, and (a better-defined) Controller is rather interesting and is much more descriptive and useful than the four class roles. If nothing else, monster roles doesn't even suffer from problems 1 and 2 as much, since their purpose is presented in a better manner and they serve as a useful descriptive tool rather than an excessive mechanical limitation. I also like that some roles, like Leader, can simply be added on top other roles, rather than be an entire role in of themselves.
4) Classes should only have a single role.
If roles are to work, then they serve as a description for a class's abilities, rather than as a prescribed, necessary part of a team. A class should be focused enough that its strengths and weaknesses (the most important part of a role) are set. At the same time, outside of those fixed strengths and weaknesses a character should be flexible, so "alternate roles" would only make sense or be necessary in a system of roles that is overly limiting or overly prescriptive.
Well, that last one is more of a collision of my preferences in both roles and class designs, but I still think it is important.