Short answer: I would say that for can be omitted, but everyone may not agree with me. Its felicity may depend on context.
My answer focuses on the question you overheard, but my comments apply also to the question in your title ('I've been doing this (for) a week').
As a native speaker (AmE), I tend to think that whether for is necessary depends on the construction, context, and current usage, as well of course on the speaker's dialect or idiolect.
I certainly would rarely say
[Can you] wait for a second/a minute, I need to tie my shoe.
I would omit for in the sentence above. This is true whether I express it as an imperative (not using can you) or a question. Ngrams seems to back me up on this:
A search using second in both phrases returns a flat line (comparatively zero results) for the expression with for.
Another construction where the omission of for seems well suited is
I'm gone ten minutes/two days and (come back) and this is what happens?
Again, a short(er) period of time seems to work better here.
They've been going out a week and this is what happens?
As for the original sentence
They've been going out a week; I mean, that's not (a) serious (relationship),
in general, it might sound better with some qualifier such as only or now:
They've been going out a week now; I mean, that's not (a) serious (relationship).
You know, a lot depends on delivery (how it is said).
If it is said with no stress on any word or meaningful pauses in delivery, it seems a bit 'iffy' to me, but I would not judge it substandard but only say that for would improve it.
However, if week is stressed, it is definitely well suited or 'felicitous'.
Likewise if there were a meaningful pause between the two clauses, as in
They've been going out a week;...I mean, that's not (a) serious (relationship),
One assumes the first clause is a statement of fact, and if it is followed by a meaningful pause (a few moments is long enough) and then the second clause is stated as a 'commentary' on the first, that sounds fine also.
Used on its own the first clause sounds especially felicitous without for in answer to the question:
How long have they been going out?
and the reply
They've been going out a week.
Note we sometimes use the grammar or construction of the question in our responses.
So if the question had been
How long have they been going out for?
(whose felicity seems questionable, or at least to depend on dialect)
the response with for would not be unexpected.
Thus, since the interrogative form seems better to me (and I would think, most native speakers) without for, I'm not surprised I can accept the like response.