Most writers use that and whichas the relative pronouns for inanimate objects, and who as the relative pronoun for humans. This widespread habit has led to the mistaken belief that using that in reference to humans is an error. In fact, while most editors prefer who for people, there is no rule saying we can’t use that, and that has been widely used in reference to people for many centuries. It remains so today, especially in British writing, exemplified here:
Labour insiders argue that Balls is probably the man that can most effectively deliver a reshaping of the party’s economic policy. [The Guardian]
This demonstrates how completely out of touch they have become with the people that they purport to speak on behalf of. [Telegraph]
The use of who where that is more appropriate is rare, but it happens on occasion, especially with reference to companies and corporations, which are obviously not human—for example:
So the goal would be to look for companies who sell products that are in demand overseas. [San Francisco Chronicle]
As for whether it’s okay to use who in reference to animals, this is a matter of preference. Some people think of their cats, for instance, as thinking beings with real personalities and wouldn’t hesitate to refer to them with who. The same people might not do the same for, say, a jellyfish. But most edited publications use that and which in reference to animals.