A Mod Chip.
No, you can't play burnt games with the Twilight Hack.
No, you can't play burnt games without a mod chip.
Yes, you have to open your Wii.
Yes, you will void the warranty on your Wii.
To find out what chip you need, you can use one of the serial number sites to guess at what chip set your drive uses. But since you need to open your Wii to pop the chip on, you might as well open it up to find out without a doubt.
There are tons of guides on how to open your Wii and see what chip set your drive uses. No one will tell you how in this thread. Look it up.
Please don't post asking which modchip you should get for your chipset, as there are already 18 bajillion threads already asking that. Please use the search.
Hope this helps!