Support them rather than abandon them
Is the above correct? Is the repeating of 'them' correct? Can we drop 'them' when we use it second time?
Yes, it is correct.
From the perspective of grammar, you can use them twice.
As for using only a single them, only the first one can be removed without it becoming ungrammatical:
✘ Support them rather than abandon.
✔ Support rather than abandon them.
Others have provided good answers, but they don't really say why you can't drop the 2nd "them" in order to be "grammatically correct."
The reason is that abandon, when used as a verb, is transitive: It requires an object.
So why can the sentence be re-worded like this?
Support rather than abandon them.
In that case, "rather than" is acting as a conjunction and "support rather than abandon" is a compound verb.
For description of "rather than" as a conjunction, see https://www.merriam-webster.com/words-at-play/usage-of-rather-than.
For description of compound verb, see http://grammar.yourdictionary.com/sentences/15-sentences-using-compound-subjects-and-compound-verbs.html
"rather than" separates two options and indicates a preference for the former. To be correct each of the alternatives should be grammatically correct in the entire phrase. Thus:
We should support them rather than abandon them.
indicates two alternatives:
We should (support them).
We should (abandon them).
Where the brackets indicate the alternatives. The other form:
We should support rather than abandon them.
indicates the alternatives:
We should (support) them.
We should (abandon) them.
Which are identical to the previous case but are derived differently.
This is the only requirement for a phrase using "rather than" to be grammatical, that is, each alternative, when used in isolation in the phrase, should be grammatical. Apart from this the alternatives need not be similar. You can say:
We should support them rather than go to the theatre and then have dinner.
although this is ambiguous. The first, favoured, alternative could be
We should support them.
or it could be
We should support them and then have dinner.
In both cases the second alternative is:
We should go to the theatre and then have dinner.
Personally, I would say "who" is being supported in the first clause and then use the object pronoun in the subordinate clause; e.g.,
Support our vets rather than abandon them
(vets = veterans or veteran soldiers)
Support them rather than abandoning them
"Rather than" cannot be used as a conjunction to join two verb phrases. If the unit following "rather than" is a verb phrase, then its head verb should be put into the "-ing" form to turn it into a noun phrase.
Randall Stewart explained why the second "them" is needed.