What role if any does academic scholarship play in Buddhism?
It's beneficial to know ones way around the Pali canon and to have knowledge about the different doctrines and suttas.
If one is a teacher then one would have to do some level of studying the canon in order to teach others what the content is. It might also be beneficial to learn how to read the Pali language in order to get the "full" meaning of the words. It's not uncommon that pali words loose part of their meaning when tranlated into other languages.
So it's important to study but that studying cannot stand alone. It must be combined with practice in meditation in order to fully understand what has been studied.
I remember hearing a story about a young monk (or maybe he wasn't yet a monk) that were described as not being very bright. He could not read and had a hard time understanding what has been taught to him. He was simple and pure. One day a teacher told him to just go and observe his mind. He did that and he won Nibbana in no time while the other hard-studying student with clever minds took a lot longer to win Nibbana because their heads were filled with intellectual knowledge.
Unfortunately i do not remember where i heard the story and i do not know if i have used the correct words to describe it. But the essence of the story is that one does not need to have a large amount of intellectual knowledge to win Nibbana. To win Nibbana practice is needed and that can be gained by simple instructions from a teacher.
Should Buddhists be concerned with what scholars say, or can we safely ignore them?
It comes down to what the goal of the Buddha's teaching is, i.e. to achieve freedom from suffering. To win Nibbana one has to practice insight meditation. Studying is simply not enough since the intellect cannot penetrate reality to it's core. Intellectual knowledge (book knowledge) will only get one so far. Experiental knowledge (insights into reality) is needed in order to fully understand the immense deepth of doctrines such as anatta.
These concepts cannot be grasped by the intellect. They are too profound. They must be experienced directly through insight meditation. Nibbana cannot be won by studying.
I like to use the onion-simile for illustration. Imagine we have an onion with it's many layers and a core. Intellectual knowledge can only penetrate the top layers of the onion while experiental knowledge can penetrate all the layers and into the core of the onion, i.e. Nibbana.
Wether or not the words of scholars should be ignored is really up to the individual being. If we take the perspective of gaining freedom from suffering then the words can be ignored if they do not have a direct relation to the decreasing of suffering, the winning of Nibbana or can be tested in insight meditation.
How should we resolve conflicts between tradition and academia?
By doing insight meditation one gets a point of reference to reality. When one has seen for oneself how reality functions then there is no need for academic discussion. Buddhism is not like philosophy where there are multiple opinions or views on a topic. We can go and test everything in insight meditation and thereby gain the right understanding of reality.
It is not like other religions where there is only faith. In Buddhism faith is balanced and guided by wisdom so that it does not become blind faith.
There are many more aspects to the questions. I have only brought up a few which i found important in the relation between Intellectual buddhism and Practical buddhism.
EDIT: The story i told above is not the correct one. Bhante has provided the true story.
It's the story of Cula-panthaka. He was a monk who could not remember a single stanza and because of that he was ordered to leave the Order. He was very sad about this and on his way out of the monastery he met the Buddha who comforted him and gave him a clean piece of cloth and said "Sit with your face to the East," and "repeat the words 'rajoha-ranam' and wipe your face with the cloth". Cula-panthaka did what the Buddha told him and wiped his face with the cloth. He saw that the cloth became dirty and then concentrated his mind on the impermanence of all things. The Buddha gave him a discourse telling about the importance of getting rid og impurities and other evils. At the end of the discourse Cupa-panthaka attained arahantship with knowledge of all the Pitakas.