In the early chapters the narrator is shown struggling with his own belief of reality--he worries he may not be keeping his grip on reality. That being said, it probably won't be a problem for people struggling with paranoia.
The mother of a main character is lied to several times so she wont worry about her daughter
(This book is very heavy on the notion that women can be hysterical)
it's implied. vampirism and the exchange of blood is a metaphor for sex throughout this novel, so there's a scene that could be interpreted as sexual assault. another character is sort of hypnotized and coerced into sexual actions with vampires
A patient in a mental institution does have a scene whilst still at the institution but it does not discuss treatments or doctors. It just happens to take place there
A character has symptoms of PTSD such as flashbacks, but since the novel was written before PTSD was an official diagnosis, it isn't explicit if they have it
A mental institution features including mentally ill patients mentioned. As it is set in the 1800s this is pretty ableist and uses some very outdated and offensive terms
Mostly not explicit, but vampirism draws on some anti-Semitic stereotypes eg "fear" of crucifixes and blood-libel (children are drained of blood).
Towards the end, the only explicitly Jewish person is an ally of Dracula and described as loving money and having a nose “like a sheep”.