If the emotions' only purpose is to stimulate certain behavior, why can't this task just as effectively, or even more effectively be done by the reasoning part of the brain?
Are emotions really necessary for survival?
It appears that there's been a lot of research done by USC professor Antonio Damasio on the importance of emotions.
Emotions are important because they end up directing reason. Without emotion, there are simply too many parts of any situation or decision for reason to know when to stop thinking and actually make a decision. Have you even been in a position where you need to choose between two alternatives that seem equally good to you (for example, deciding which of two equally-qualified candidates to hire), and you spend what seems like an eternity weighing the pros and cons of each, coming up with new reasons and rationales, before reluctantly making a decision? That's because you didn't have emotions pushing you one way or the other.
You might think, wait, why wouldn't we be able to make decisions quickly without emotions? You'd be able to -- people with impaired emotions have no problem making quick decisions when directed to do so. Without such direction, they have no reason to make decisions quickly because there's no emotion telling them that making decisions quickly is "good" -- in fact, there's nothing telling them that anything is "good".
That's a highly-simplified summary of Dr. Damasio's findings. I suggest reading more about his work if you are really interested.
This question becomes more complicated if we think in terms of "emotions" (e.g., angry, happy, sad, afraid, etc.) than in terms of "affect" (positive and negative feelings, high and low arousal). I'll start with affect and move on to emotions.
An affective state tags an object with a certain value--and it does so very quickly (e.g., Pham, 2007). For example, if a snake makes you feel bad, then it must be undesirable. This motivates a plan of action that could involve elimination of the snake (fight) or withdrawal from it (flight). You do not have to spend time considering the value of the snake (i.e., that it's bad and dangerous) and developing a plan of action to deal with it. Instead, your negative affect facilitates automatic evaluation and action, which might increase your evolutionary fitness. Gerald Clore has suggested that negative affect serves a STOP function whereas positive affect serves a GO function.
Emotions are a bit more complicated, in part because we don't have a consensual definition of them. If emotions are merely socially shared concepts (happy, sad, angry) that we use to describe and organize our situated affective experiences (e.g., Barrett, 2014), then we can say that emotion concepts serve self-regulatory and social-communicative functions. For example, by labeling as anger my high arousal and negative affect in the context of having my autonomy violated, I've organized this multifaceted affective experience into a single concept (of anger). This enhances my capacity to develop ways to regulate my affective experience (Kashdan, Barrett, & McKnight, 2015) and it facilitates communication of it. If I tell you that I am angry, then you'll have a general idea of what I'm feeling. Self-regulation and social communication are certainly important for survival.
Emotions are involved in lots of other things, e.g., facilitating cooperation (e.g., DeSteno et al., 2010), motivating the pursuit of new goals (Bench & Lench, 2013), social bonding, deceptive behavior, cheating, and so on. All of these can be adaptive in certain contexts.
It's also important to note that emotions are flexible. For example, we don't have a "fear module" in the brain that motivates a rigid pattern of action (e.g., Lindquist et al., 2012). Instead, emotions (on one view) emerge from the interactions of more domain-general processes (including exteroceptive perception, language, interoception, conceptualization, executive attention, and core affect; Oosterwijk, Touroutoglou, & Lindquist, 2015). These domain-general processes interact to produce emotional episodes with patterns of cognition, behavior, physiology, and experience that are flexible to the situation at hand (Barrett, 2015). The kind of flexibility and efficiency that you might expect from the "reasoning part of the brain" probably does not match up to the kind of flexibility that I just described. Also, it's important to note that emotion and cognition are not separable constructs.
EDIT: As far as reason vs. emotion is concerned, emotion is a much more potent motivator. If you consider seizure patients with lesions to the majority of their amygdalae (from lobectomies), they don't show fear. If you put them in a dangerous situation, they won't run away--despite having their prefrontal areas intact. (Although an alternative non-emotional interpretation is that these patients have lost the ability to discern what is salient, e.g., threatening, in their environment.) If you have a hobby that you really enjoy, your motivation to continue doing it is largely emotional ("It makes me happy"). As onigame implied in his/her answer, emotions motivate these kinds of decisions.
Anyway, I think I've provided you a very incomplete view on the adaptiveness of emotions, but hopefully this clarifies some things.
Are emotions really necessary for survival? No, not for survival; lots of living things around us without even a brain.
Did emotions provide an evolutionary advantage in the past? The areas of the brain we associate with emotion were around far back in our evolutionary past - long before conscious reasoning appeared. Emotions are still an important part of reasoned decision-making in humans, but for our early ancestors, they would have been crucial.
Are emotions now "vestigial" (relics of the past) holding us back from making decisions more effectively with reasoning alone? Unconscious decision-making is far faster, works in parallel, high capacity, and is remarkably effective, while rational decision-making is slow, sequential, low capacity, and bounded in effectiveness. So as it stands, removing emotion from decision-making would cripple us.
But given that we now have the capacity for rational thought, don't the disadvantages outweigh the advantages? Disadvantages of the emotional system include failing to use reasoning effectively at times when it is practical and better to do so, the effect of emotional (mood) disorders, and loss of self-control in emotionally intense situations. Evolutionary psychologists have long argued that systems that provided an evolutionary advantage in our past (such as 'stress') may no longer be as advantageous in modern society.
For completeness, I should mention that for a lay-person, the term 'emotion' may refer to a number of distinct terms in cognitive science. Modern theories of emotion suggest that like many aspects of self-knowledge, emotion is "inferred" - rather than "introspected" - from a combination of physiological cues and other context information. That is, emotion is a person's interpretation of their cognitive state, so the idea of 'removing' it is not well defined.
'Feeling' is the phenomenological aspect of emotion - the actual conscious experience of it. One might sometimes wish that we could make decisions based on emotional state without the experience of it... For example, sometimes pain causes me to stop doing something that might injure me, but other times, I stop doing it but the pain persists, and it sure would be nice if that (apparently useless) pain stopped. Hence the popularity of pain killers, anti-depressants, anti-anxiety, and other emotion-suppressing medication. As cognitive scientists don't really understand conscious experience, it is not clear whether or not 'feeling' could ever be removed from 'emotion', allowing us to make effective decisions without unpleasant experiences.
'Emotional expression' is the externally observable aspect of emotional state. Emotions may play a significant role in all decision-making, but this isn't always apparent externally due to individual differences in expression. The evolutionary advantage of emotional expression (or lack thereof / self-control) is largely social.
Emotions are not necessary for survival, but they may provide evolutionary advantages.
Although crocodiles cry, they do not feel any remorse in killing their prey when shedding tears doing so (they empty their lachrymals when snapping their jaws shut). Given their relative brain size we can expect (but never know) that they do not experience a comparable range of complex emotional states as we humans do. Just as fishes and amphibians crocodiles are though to be pretty much instinct-driven creatures. However, admittedly, when one sees a mother croc carrying her hatchlings to the waterside you start wondering...
Source: Ye-Tao et al., 2010
Nonetheless, assuming they are driven by instinct and are emotionless (or at least relatively so), fishes and reptiles have been around for ages and are still thriving to date. In the end, we evolved from a fishy ancestor.
Your question then becomes a more subtle one, as emotions may have given us evolutionary advantage over other speciesm but it is not obligatory for survival. Disregarding the fact whether we are better or worse than other species of animal, we do dominate the world, for the better or for the worse. I think emotions have aided us in this domination and have given us an advantage. We tend to care for others even if we don't see them anymore (emigrated working class folks that keep on sending money back home), or even when we don't know them (we donate money to other people in foreign countries out of empathy). Emotions makes us protect our kids (don't make mummy angry or it will spawn the lioness) which makes the parents more evolutionary "fit". Hence, emotions are not obligatory for survival, but they do give an evolutionary advantage.
Ye-Tao et al. J Accord Int Med;6:43-58 (only for the image).