i) Some atheists as well as some Muslims say the Bible endorses honor-killings. The allegation is based on equivocal definitions. In Islam, if a female is raped, then she has brought dishonor on her kin. It's not the rapist who dishonored her or her kin. Rather, the rape victim dishonored her kin by allowing herself to be raped (not that she could stop it.) Therefore, it is not the rapist who is punished, but the rape victim.
Needless to say, the Bible doesn't begin to endorse honor-killings in that sense.
ii) Now, there is a sense in which the Bible contains honor-killings, viz. Gen 34 & 2 Sam 13. These, however, are very different from honor-killings in the Muslim sense:
a) It is not the rape victim, but the rapist and/or his male kinfolk who is (are) punished. In the case of Gen 34, the rapist's male relatives are considered complicit in the crime.
b) These are honor-killings in the sense of revenge killings. Male relatives of the rape victim avenge their dishonored relative by exacting retribution on the perpetrator and/or his clan.
So (a) & (b) operate with an honor code that's the polar opposite of Muslim honor-killings.
iii) These are Biblical examples rather than Biblical commands. Descriptive rather than proscriptive.
iv) I've read four commentators (Arnold, Chisholm, Vannoy, Youngblood) who classify Absolom's action as "murder." However, the account itself doesn't characterize his action as murder.
Arnold goes so far as to say his action was "totally unjustified." But this seems to be a case of commentators projecting their scruples onto the text rather than deriving that value-judgment from the text.
The Mosaic Law made provision for the avenger of blood–although that meant avenging a murderer rather than a rapist. But the point is, the Mosaic Law didn't treat a revenge-killing as ipso facto illicit.
At a minimum, Absolom's action is mitigated by David's dereliction. Because David did nothing to punish the perpetrator, it was up to Absolom to defend his sister's honor.
From an OT perspective, it's not clear to me that that's murderous or even culpable. At the very least, there were extenuating circumstances. In fact, I can imagine ancient readers who'd consider it dishonorable if Absolom didn't avenge the crime committed against his sister. Certainly David's failure in that regard is dishonorable.