There is no menu entry in Nokia 8110 4G to switch to developer mode. How can you enable it without hacking firmware?
There is a special key code to toggle developer mode in Nokia 8110 4G. Just dial *#*#33284#*#* to enable or disable developer mode. The number 33284 can be dialled by typing the phone word "debug" on your phone.
The code is not saved in the dial history. That means you would need to type it from scratch each time you use it. As a short cut you can store it in the address book. Unfortunately you can not dial it using the address book entry directly. But when you start typing *#*# the address book automatically suggests this entry and you do not need to type the rest of it.