I block or deny access based on the host name or IP address of the client visiting website under nginx web server. I want to display customized e403.html error page, but it doesn't appear to be working. Nginx always displays the built-in, hardcoded "403 Forbidden" error message. Here is my configuration:

.....
  ...
 deny 1.2.3.4;
 deny 91.212.45.0/24;
 deny 91.212.65.0/24;
 .....
 ...
 error_page 403 /e403.html;
        location = /e403.html {
           root   html;
       }

How do I fix this problem and display custom error 403 page under nginx server?

The deny parameter will block all access including access to /e403.html file. All you need to add is allow all; inside location directive as follows. Edit /usr/local/etc/nginx/nginx.conf or /etc/nginx/nginx.conf, enter:
# vi /usr/local/etc/nginx/nginx.conf
Update it as follows:

 error_page 403 /e403.html;
        location = /e403.html {
           root   html;
           allow all;
       }

The above should fix the problem. Once done, reload the nginx web server:
# /usr/local/nginx/sbin/nginx -t && # /usr/local/nginx/sbin/nginx -s reload