*She told us [to read the text carefully], and [that we should write a short review of it].
The OP's example has two complements to the verb head; the complements are bracketed. In such a construction, I wouldn't use a comma:
(1) She told us [[to read the text carefully] and [that we should write a short review of it]].
This construction is still not very good; a different construction using parallel structure would be better:
(1a) She told us [[to read the text carefully] and [to write a short review of it]].
Back to the original example, placing a comma would be fine if we remove the complementiser that:
(2) She told us to read the text carefully, and we should write a short review of it.
Here we have two independent clauses joined by a comma and a coordinating conjunction. (Whereas in the original example, the declarative content clause that we should write a short review of it is not independent.) This, however, is not as good as (1a).
If the verb head has three complements, instead of two, using commas would be fine:
(3) She told us [to read the text carefully], [to discuss with team members], and [to write a short review of it].
Here I have used parallel structure, similar to what I explained in (1a).
The following would be fine too:
(3a) She told us to [read the text carefully], [discuss with team members], and [write a short review of it].
The Oxford comma is optional.
Edit
The second clause of my (2) is detached from the teacher's instruction, and hence the sentence is not equivalent to the OP’s.