Is the tense of the latter part of the sentence correct?

Couldn't is the past tense of 'can't'; the situation being described is in the past; the burning down was a hypothetical possible consequence; the correct verb form for 'burn' is the future in the past (would burn): he couldn't foresee devoting himself to anything other than that passionate desire, even if the fire would eventually burn him down."

Future in past

You have two options here: backshift both clauses, or keep the original tense.

Assuming the original utterance was:

I can't forsee devoting myself to anything other than that passionate desire, even if the fire eventually burns me down.

Then you can backshift:

He said that...

he couldn't forsee...fire eventually burnt...

or not:

he can't forsee...fire eventually burns...

Both leave it open as to whether he is burnt down by the fire in the end. The outcome can clearly be questioned in both cases.

The addition of a modal auxiliary verb just adds modal meaning to the sentence that wasn't there in the first place.