What is the role of thinking in Buddhism?

Again, not speaking for the whole of Buddhism, but from the perspective of Rinzai Zen, conceptual thinking isn't so much of a problem in that it gets in the way. It's more of an impediment because it can result in a case of mistaken identity.

Imagine you walk into an antique store with all of the lights turned off. In the pitch black, you are asked to get a sense of all the items in the room. You walk up to each shelf and pick up each item one by one. You feel their weight, their delicacy. You become intimately familiar with the materials they are made of, their structure, whether they feel warm to the touch or cold. If you were to walk into the the same antique store with all of the lights turned on, your inclination would be to rely exclusively on your sense of sight. You probably wouldn't even bother to pick anything up at all!

In Zen, the antiques are koans and the darkness is the calm and empty mind of mushin. If you were to rely on your discursive mind - your sense of sight in the metaphor above - any conclusions you arrive at would be superficial. All of your spiritual senses just aren't engaged in the same way as they are when you are in mushin. All too often, people think that they understand a koan simply because it makes intellectual sense. They don't push themselves past that point because they think that sort of understanding is all that's required. Instead, what they really should be doing in exploring that koan thoroughly - not just with their minds but even with their whole bodies.

Zen is about developing intimacy with the world through a mind marked by emptiness. Conceptual thought can sometimes distance us from what is really going on in the universe. We all too quickly assume that what we see, hear, or learn is the end of the story. It's not. For true understanding to occur, we need to get our hands dirty.

This is, as you say, a hot topic of debate, but based on my study of the Pali suttas and the original writings of Padmasambhava, my own view is that the Buddha's teaching is that wisdom is the essental salvific principle, and that thinking is part of this process. The Buddha definitely taught that one can think about dharma, while at the same time denying that thinking about dharma alone is sufficient for emancipation. Rather, thinking ultimately leads to its own transcendence, but this can only be achieved as a result of thinking. The Zen view can lead to error and even delusion, especially in a non-Buddhist society. Remember that Zen appeared in societies that were already steeped in dharma. Also, I think that the view that Zen eschews thinking is facile. For example, the Sutra of Perfect Enlighenment is a Zen text, and is clearly deeply philosophical.

This is a question related to Lo-Rig, the study on valid cognition, 'epistemology'. An authoritative and simple text on the topic is Mind in Tibetan Buddhism, by Lati Rinpoche.

To answer in a practice-oriented fashion: when one is subject to anger (a conceptual consciousness that is also a wrong consciousness in that it engages its object erroneously), one comes to tell himself stories about the object. For instance, one comes to think “This person is detestable; how could have I been fooled and even like her? How could anyone love her? If they knew her for who she is, they would understand; she really wronged me...”. Finding all the more reasons, if not to be angry, at least to justify one's anger. As practitioners we engage in analytical meditation on impermanence & death, karma, refuge, precious human rebirth, and so forth, on a daily basis. It is called “Meditation on the Lam Rim”. We do it for two reasons:

1. So as to tell ourselves factually concordant “stories” in order to generate, for instance, patience and dwell in there, familiarizing with being patient, and so forth
2. So as to be able to pay attention to facts that we are so often blind to. For instance, after meditating a great deal on buddha-nature, it will be easier to find / see reasons not to despise any sentient beings.

Even when engaging in Calm Abiding, one needs to alternate between (1) analytical meditation and (2) placement, because one has to analyze the object he means to achieve Calm Abiding with in order to (1) familiarize with the object and (2) cultivate the clarity factor with regard to it.

Once our eyes are fully open, once we see reality, there is no need for these anymore. But in our (Geluk, Madhyamika-Prasangika) presentation, only a buddha is free from [the need for] conceptual consciousnesses, and is therefore called 'a valid person'. Also, since even direct perceivers in the continuum of a sentient being are not free from the appearance of true existence, abandoning conception [even the conception of true existence] is not sufficient. That is a unique Tenet of the Prasangika school, though.

To answer in a more technical fashion: a conceptual consciousnesses apprehends its object by way of a mental image (also called 'meaning-generality' or 'generally characterized phenomena'). All conceptual consciousnesses are mistaken [with regard to their appearing object, a mental image] in that they mistake the appearing object for the object of engagement. However, a conceptual consciousness, even mistaken, can be valid. For instance, an inferential cognizer (which is a conceptual) is always valid because it necessarily arises in dependence upon a correct sign that has been realized [and which pervasion in relation to the predicate as well]. This accords with Purbuchok's presentation (in Explanation of the Presentation of Objects and Object-Possessors as well as Awarenesses and Knowers).

We often present the gradual process of generating valid consciousness as follows: Having doubts turning away from the factual [in relation to a slightly hidden phenomena, such as for instance subtle impermanence], one listens to the teachings – listening also comprehends reading – and comes to generate equal doubt. He keeps on listening to the teachings and comes to generate doubt tending towards the factual. He keeps on listening to the teachings, gives it a thought, and comes to generate a correct assumption. Most positions we hold as true without being able to establish by way of reasoning are such. Doubt and correct assumptions are not 'realizing consciousnesses'. Therefore, strictly speaking, someone having generated a correct assumption with regard to a slightly hidden phenomena has not 'realized' anything yet. From there, he engages in reflection and/or in meditation in an analytical fashion and, in dependence on a correct reason, comes to generate an inferential cognizer. That is why we engage so much in debates, familiarizing with reasoning itself, but also with reasons and their pervading such and such predicates (and therefore establishing thesis). An inferential cognizer i is the first realizing (and valid) consciousness in this scenario. It is valid because it engages its object correctly, but it is mistaken with regard to its appearing object, a mental image, because it is a conceptual consciousness. Although a buddha is free from inferential cognition, [valid] inferential cognizers realizing the sixteen aspects of the four noble truths, and so forth are to be cultivated on all the path of learning. It is taught that a slightly hidden phenomena – such as subtle impermanence, and so forth – can be realized directly after having familiarized realizing it inferentially. For instance, subtle impermanence or emptiness are directly realized for the first time in the context of a meditative equipoise (which is a union of calm abiding and special insight) on the path of seeing. Such a realization is preceded by several instances of inferential realization, that is conceptual cognition, also being unions of calm abiding and special insight. A wisdom directly realizing emptiness is not conceptual but is analytical because it is a union of calm abiding and special insight, and special insight has the function of analysis. Only a buddha is free from conceptual consciousnesses.

Wrong consciousnesses (ex.: doubt turning away from the fact, afflictions, etc.), non-realizing conscioussnesses (such as wrong consciousnesses, all types of doubts, inattentive awarenesses, etc.) mistaken consciousnesses (conceptual consciousnesses, including [valid] inferential cognizers) are objects of abandonment. However, they are not to be abandoned by any practitioner. The process of abandonment is gradual... and that which is to be cultivated by one is to be abandoned by another. For instance, inferential cognizers are not to be abandoned by those who still have afflictive obscurations.