Is it permissible for a doctor to refrain from treating "cold cases" for those they are not responsible for, in order to avoid problems, while still treating emergency cases?

If the institution's system only allows the disbursement of treatment to a specific category, then it must be adhered to. It is not permissible for an employee to resort to lying if they fear harm, as long as they are able to use petitions to avoid it. However, if they cannot avert harm from themselves except by lying, then it is permissible for them to do so out of necessity. The limit of necessity is when it is highly probable that one will fall into perdition because of it, or that they will suffer unbearable hardship because of it.