What could go wrong if one misconstrue Buddhism with nihilism?

I've no formal experience with borderline personality disorder, this answer isn't intended to be relevant to that.

I've seen posts on this site where people say, "If you don't believe in rebirth then you believe that death ends everything (and ends suffering). But if you really believed that death ends then why wouldn't you just kill yourself? Therefore you must believe in rebirth." I think that (i.e. the belief that death ends suffering) is one of what Buddhism identifies as "a wrong view" and might be translated as nihilism (or "annihilationism"). There's even a sutta in which some monks do kill themselves and the Buddha clarifies that that wasn't right.

Conversely I think that the (non-Buddhist) Bhagavad Gita too seems like a wrong view to me (at least as far as I understand it). I think there's a scene where someone is faced with 'enemies' before a battle, and he doesn't want to kill them because they're his kin, and the God says to him something like, "But they're going to die anyway, everyone does, they're mortal[s]. Now you should go and fight (and kill) them because that's your social duty, the job you were born to do (by being born into the warrior or ruling class, I guess), and therefore your holy duty." The Buddhist doctrines of emptiness and anatta notwithstanding, I think that "There is no father, no mother, no teachers" and so on is explicitly declared to be another wrong view.

There's a hint of that in the Zen story which is titled, Nothing Exists.

That's about it, I'm not sure what else would count as nihilism. Possibly despair, a feeling of helpless -- or a feeling of heedlessness -- a view like, "It's all impermanent, nothing makes any difference, no point in even trying anything."

I read the link you posted, e.g. ...

Frantic efforts to avoid real or imagined abandonment by friends and family

... but, sorry, I'm not sure how to advise about that or even what it has to do with nihilism exactly. I think I see it's related to psychological "attachment theory". The Buddhist doctrine I've learned doesn't seem to me to be very much about such relationships, except things like this and doctrine about the brahmaviharas. And the "refuges", the three jewels, suggest maybe people shouldn't put their faith in friends and family and so on.

On the other hand something like DN 31 says there is such a thing as proper behaviour for a layperson, and there is a difference between a good friend and a bad friend, that there are duties between between family members (and so perhaps to that extent, "good" and "bad" family members), maybe that's worth knowing too.

... so they want to limit relationships at all cost to protect the person involving them.

I suppose it's because a therapist isn't too attached, and because they're motivated to "first, behave ethically (and/or 'professionally')", and is "disinterested" i.e. needs no help from the patient, but not "uninterested" so there is two-way communication, I think those are reasons why a a patient needn't fear that they may hurt and therefore must protect their therapist (though shying away might be a habit that's hard to break, and sharing personal information might be a skill like any other which if it hasn't been learned already might be difficult to practice).

Instead of being entangled (in a relationship), protecting a partner or protecting oneself from a partner, perhaps with a therapist it's clearer (to the patient) that it's the patient who will "inherit their own kamma":

I am the owner of my actions (kamma), heir to my actions, born of my actions, related through my actions, and have my actions as my arbitrator. Whatever I do, for good or for evil, to that will I fall heir...

I think that that doctrine -- that reflection or observation -- is used, to say that it matters what you do and what you intend: because that actually makes a difference (to you, to your future self, and/or to whoever may exist in the future).

I think that Buddhism also uses this doctrine to teach some equanimity towards others (e.g. that "they are heir to their own kamma") -- the brahmaviharas -- i.e. you're supposed to feel kindness and compassion (maybe being harmless and helpful, not hurtful), and admiration (for someone's virtues, instead of being jealous or pitying), but also equanimity (disentangling).

The bit about heedlessness is what I'm talking about. Can you elaborate?

I think that the "there's nothing I can do that makes a difference" view is a wrong view, perhaps canonically associated with a view that "what happens is predetermined or predestined" -- possibly a clockwork universe theory -- i.e. that past kamma determines everything about "now", and will also continue to determine everything about the future, and so there's no sense in trying and no personal responsibility for anything.

I think the Buddhist view is that, in contrast, a) kamma is intentional action; b) past kamma does affect the present but isn't the only thing that affects the present (e.g. a hurricane isn't a result of kamma, though how you react to a hurricane might well be); c) what you do now (and what you intend now, i.e. new kamma) does make a difference -- whether you intend good, intend evil, or even intend to "end" kamma.

Also if the "clockwork universe" theory is about physics, Buddhism is about the mind, for example:

1. Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with an impure mind a person speaks or acts suffering follows him like the wheel that follows the foot of the ox.
2. Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with a pure mind a person speaks or acts happiness follows him like his never-departing shadow.
3. "He abused me, he struck me, he overpowered me, he robbed me." Those who harbor such thoughts do not still their hatred.
4. "He abused me, he struck me, he overpowered me, he robbed me." Those who do not harbor such thoughts still their hatred.
5. Hatred is never appeased by hatred in this world. By non-hatred alone is hatred appeased. This is a law eternal.
6. There are those who do not realize that one day we all must die. But those who do realize this settle their quarrels.

Being "heedful" in't easy. Canonically it was the Buddhas final advice, the last words of the Buddha:

vayadhammā saṅkhārā appamādena sampādethā

The opposite of "heedfulness" might be likened to a "drunkenness".

I say "not easy", maybe it's like that famous bit of advice to Christians, "be perfect".

Anyway I think that being "heedful" may be related to being "mindful", some ability to concentrate, or if not "concentrated" then to not get carried away -- to remain in the present.

Incidentally the classifications in the DSM are quite approximate:

• It says e.g. "this not that" where really people are a bit of both
• Also "this" might be a characteristic of relatively healthy people too, what makes it unhealthy is a matter of degree (doing it "excessively"), where the definition of "excessive" is difficult to assess if you're, like, not professionally trained, and to some extent it's a matter of personal choice.

Anyway, apart from "BPD" there's another diagnosis, "PTSD", which for all I know (i.e. little) might be similar in one way -- i.e. it's where someone has (had) a traumatic experience and keeps "re-living" it. One of the psycho-therapeutic therapies for that is what they call "mindfulness" (that's probably not exactly what the Buddhist canon defines as "mindfulness", but still), especially trying to be mindful of or to remember the present (instead of re-living the past).

It's "maladaptive" behaviour e.g. it's fearing a threat where there isn't (or shouldn't be) one in the present.

Perhaps that's some form of "attachment" but an attachment to a past undesirable event.

An actually that might be another side to "impermanence" i.e. sometimes it's difficult to get that some past threat, a threatening situation, is over and finished, that the present situation is different, is changing.

I imagine there are lot of ways in which Buddhist doctrinal views could be applied to psychotherapy, or vice versa, maybe I see some parallels (but I haven't experienced that, not been trained).