I think we there is no point in creating a new input stream if the response type is `EMPTY`.

You're right.

Moreover, shouldn't we call handleEmptyEntity() in that case?

No, JFrogServiceResponseType.EMPTY means that the service expects nothing to be returned (void) and supposed return successfully, whereas handleEmptyEntity handles a service that expects to find one but got no entity.
@yahavi, I propose the following:

private void processResponse(HttpEntity entity) throws IOException {
    if (responseType == JFrogServiceResponseType.EMPTY) {
        return;
    }
    if (entity == null ) {
        handleEmptyEntity();
    }
    try (InputStream stream = entity.getContent()) {
        setResponse(stream);
    }
}

@Or-Geva OK, but please add Javadoc above handleEmptyEntity explaining when it should and shouldn't be called.