For a strong AI, the short answer is to call for help, when they might not even know what the supposed help could be.
It depends on what the AI would do. If it is supposed to solve a single easy task perfectly and professionally, sure emotions would not be very useful. But if it is supposed to learn random new things, there would be a point that it encounters something it cannot handle.
In Lee Sedol vs AlphaGo match 4, some pro who has said computer doesn't have emotions previously, commented that maybe AlphaGo has emotions too, and stronger than human. In this case, we know that AlphaGo's crazy behavior isn't caused by some deliberately added things called "emotions", but a flaw in the algorithm. But it behaves exactly like it has panicked.
If this happens a lot for an AI. There might be advantages if it could know this itself and think twice if it happens. If AlphaGo could detect the problem and change its strategy, it might play better, or worse. It's not unlikely to play worse if it didn't do any computations for other approaches at all. In case it would play worse, we might say it suffers from having "emotions", and this might be the reason some people think having emotions could be a flaw of human beings. But that wouldn't be the true cause of the problem. The true cause is it just doesn't know any approaches to guarantee winning, and the change in strategy is only a try to fix the problem. Commentators thinks there are better ways (which also don't guarantee winning but had more chance), but its algorithm isn't capable to find out in this situation. Even for human, the solution to anything related to emotion is unlikely to remove emotions, but some training to make sure you understand the situation enough to act calmly.
Then someone has to argue about whether this is a kind of emotion or not. We usually don't say small insects have human-like emotions, because we don't understand them and are unwilling to help them. But it's easy to know some of them could panic in desperate situations, just like AlphaGo did. I'd say these reactions are based on the same logic, and they are at least the reason why human-like emotions could be potentially useful. They are just not expressed in human-understandable ways, as they didn't intend to call a human for help.
If they tries to understand their own behavior, or call someone else for help, it might be good to be exactly human-like. Some pets can sense human emotions and express human-understandable emotion to some degree. The purpose is to interact with humans. They evolved to have this ability because they needed it at some point. It's likely a full strong AI would need it too. Also note that, the opposite of having full emotions might be becoming crazy.
It is probably a quick way to lose any trust if someone just implement emotions imitating humans with little understanding right away in the first generations, though.
But is there any purposes for them to have emotions before someone wanted a strong AI? I'd say no, there isn't any inherent reasons that they must have emotions. But inevitably someone will want to implement imitated emotions anyway. Whether "we" need them to have emotions is just nonsense.
The fact is even some programs without any intelligence contained some "emotional" elements in their user interfaces. They may look unprofessional, but not every task needs professionality so they could be perfectly acceptable. They are just like the emotions in musics and arts. Someone will design their weak AI in this way too. But they are not really the AIs' emotions, but their creators'. If you feel better or worse because of their emotions, you won't treat individul AIs so differently, but this model or brand as a whole.
Alternatively someone could plant some personallities like in a role-playing game there. Again, there isn't a reason they must have that, but inevitably someone will do it, because they obviously had some market when a role-playing game does.
In either cases, the emotions don't really originate from the AI itself. And it would be easy to implement, because a human won't expect them to be exactly like a human, but tries to understand what they intended to mean. It could be much easier to accept these emotions realizing this.
Aspects of emotions
Sorry about posting some original research here. I made a list of emotions in 2012 and from which I see 4 aspects of emotions. If they are all implemented, I'd say they are exactly the same emotions as of humans. They don't seem real if only some of them are implemented, but that doesn't mean they are completely wrong.
- The reason, or the original logical problem that the AI cannot solve. AlphaGo already had the reason, but nothing else. If I have to make an accurate definition, I'd say it's the state that multiple equally important heuristics disagreeing with each other.
- The context, or which part of the current approach is considered not working well and should probably be replaced. This distinguishes sadness-related, worry-related and passionate-related.
- The current state, or whether it feels leading, or whether its belief or the fact is supposed to turn bad first (or was bad all along) if things go wrong. This distinguishes sadness-related, love-related and proud-related.
- The plan or request. I suppose some domesticated pets already had this. And I suppose these had some fixed patterns which is not too difficult to have. Even arts can contain them easily. Unlike the reasons, these are not likely inherent in any algorithms, and multiple of them can appear together.
- Who supposedly had the responsibility if nothing is changed by the emotion. This distinguishes curiosity, rage and sadness.
- What is the supposed plan if nothing is changed by the emotion. This distinguishes disappointment, sadness and surprise.
- The source. Without context, even a human cannot reliably tell someone is crying for being moved or thankful, or smiling for some kind of embarrassment. In most other cases there aren't even words describing them. It doesn't make that much difference if an AI doesn't distinguish or show this specially. It's likely they would learn these automatically (and inaccurately as a human) at the point they could learn to understand human languages.
- The measurements, such as how urgent or important the problem is, or even how likely the emotions are true. I'd say it cannot be implemented in the AI. Humans don't need to respect them even if they are exactly like humans. But humans will learn how to understand an AI if that really matters, even if they are not like humans at all. In fact, I feel that some of the extremely weak emotions (such as thinking something is too stupid and boring that you don't know how to comment) exist almost exclusively in emoticons, where someone intend to show you exactly this emotion, and hardly noticeable in real life or any complex scenerios. I supposed this could also be the case in the beginning for AIs. In the worst case, they are firstly conventionally known as "emotions" since emoticons works in these cases, so it's easier to group them together, but very few people seriously think they are, just like the example I gave.
So when strong AIs become possible, none of these would be unreachable, though there might be a lot of work to make the connections. So I'd say if there would be the need for strong AIs, they absolutely would have emotions.