What role do (ethical) values have in science according to Polanyi and Kuhn? How does Polanyi define tacit knowledge and what view would Popper and Kuhn have on tacit knowledge?
Kuhn and Popper are clearly built upon Polanyi. This is part of what makes pure rationalizations of Popper, or reductions of Kuhn to a simple post-modernism disingenuous.
For Kuhn, in particular, tacit knowledge is what validates candidates for paradigms, and what allows definitions to coalesce. The stated forms of these things are incommensurable, but we use something outside of explicit metrics to make these decisions -- they get made.
At the same time, that tacit knowledge is not entirely subjective, or paradigmatic wars would simply go on forever. Part of it is political, part of it is tradition and convention, but part of it is real knowledge that just lies at a level of subtlety below language. There are things that scientists know and do not communicate that is embedded in common experience and not in more explicit social factors.
This is equally true in all other domains, and it makes explicit rationalism into a form of authoritarian religion, or totalitarian dictatorship.