What do you mean by logging the code? On bitfinex 2fa settings page I can see a QR code and an "Accont token(key)" which is a string. Do you mean that I can for example print this page, and when I loose my phone next time I will download a google authenticator app on my new phone, feed it with the QR code that I printed and It will work as the old one?
A QR code is just a blob of data in a convenient barcode form. Your phone stores that data somewhere and uses it as a seed value (some manner of hashing operation applied to the seed and the current time produces the 6 digit code), while the server holds onto the same data to independently generate the code so that it can check them.
tl;dr : yes, the QR code remains "good" forever, and can be printed on paper or stored in a screenshot (and then encrypted) for later re-use if your phone dies.
It
is possible to access the memory where it's stored on your phone, to take a backup after the fact, but it involves rooting your phone and it's generally easier to just disable 2FA then re-enable it to take a copy of the new QR code.