Short answer: yes.
Long answer:
Which/Who/Whom/That/Nothing
Which: is used to refer to a thing mentioned in the previous clause.
Who: is used to refer to a person mentioned in the previous clause and that is the subject of the new clause.
Whom: is used to refer to a person mentioned in the previous clause but who isn't the subject of the new clause.
That: is used to refer to a thing OR a person mentioned in the previous clause.
Nothing: can replace either which, whom or that if it is not the subject of the new clause, and if it is not a non-restrictive clause.
For example, you can replace that/which in:
We have to account for every penny that/which we spend on business trips.
We have to account for every penny we spend on business trips. -> okay
But you can't here:
She healed my hand which was hurt.
She healed my hand was hurt -> wrong
Because which replaces my hand, the subject of the second clause.
So I can use "that" all the time, since it can both replace a thing and a person?
Almost. Non-restrictive clauses are clauses that are introduced with a comma, and that are an optional addition to the sentence. In these, you can't replace who/which with that.
For example:
I bought a new dress, which I will be wearing to Jo's party. [non-restrictive]
I bought a new dress, that I will be wearing to Jo's party -> wrong
I bought a new dress, I will be wearing to Jo's party -> wrong
But you can turn it into a restrictive clause with the same meaning:
I was wearing the dress which I bought to wear to Jo's party. [restrictive]
I was wearing the dress that I bought to wear to Jo's party -> okay
I was wearing the dress I bought to wear to Jo's party -> okay
For more read: http://www.oxforddictionaries.com/words/that-or-which