Checkbox Formula - field does not contain text

you can do this

If(not Contains(ToText([Approvals Needed:]),"Sales"),true)

The not just reverses the result form True to false or false to true.

But  for a Boolean result, you don't need the IF

For example, your original 

 If(Contains(ToText([Approvals Needed:]),"Quality Assurance") and Contains([Approvals Logged],"Quality Assurance"),true)

could also be written


Contains(ToText([Approvals Needed:]),"Quality Assurance")
 and
Contains([Approvals Logged],"Quality Assurance")

just easier to read 

That is basically a statement you are making and QuickBase decides if you are telling the truth and return true or else calls you a liar and returns false.

So on your second test

not Contains(ToText([Approvals Needed:]),"Sales")

will be true if the approvals do not need sales approval.