Which is correct:
We would like to thank all of those that believed in this project.
or:
We would like to thank all those that believed in this project.
Which is correct:
We would like to thank all of those that believed in this project.
or:
We would like to thank all those that believed in this project.
In the case of "all of those" vs. "all those", it's a matter of personal preference. Personally, I would use "all those"; I think it flows better.
This Google Ngram also shows preference for "all those":
I included both "that" and "who", but as others have said, "who" should be used (i.e. it's not strictly incorrect to use "that") when referring to people or groups thereof.
As was already pointed out in the comments, either is correct, but who should be preferred to that. If you are actually writing something that uses this phrase I would use a shorter version something along the lines of: "We thank all those who believed in this project". "We would like to thank" makes me wonder why you don't just thank them and makes me think along the lines of "we would like to thank, but can't" ;-) Of course that might be your intent <shrug>.
Rule of thumb: drop of before possessive pronouns its, my, our, his, her, and their and the relative pronoun that (Here is a list of all his accomplishments). Drop of before collective and plural nouns (Look at all the people in the stadium).
Use of before singular nouns (Did he eat all of the cake?), the impersonal pronoun it (Yes, he ate all of it), and objective case pronouns me, you, us, them, whom (Do all of you understand?).
From what I read, the of should be avoided whenever possible.