Is is the correct form here; in this case, the time in the subordinate clause "that everything is fine" is interpreted relative to the time in the matrix clause "I will feel that […]". The same construction even allows the past tense, as in this example:
Subjective experiences of relative success or failure flow from these evaluations, viewed in the light of one's prior expectations (e.g., among students who earn a B, any who expected a lower grade will feel successful but any who expected an A will feel that they failed to accomplish what they should have). [link; italics in original, boldface mine]
where failed is in the past tense to indicate that the (perceived) failure predates the feeling of it. (The present perfect, have failed, would also work here, but the simple present, fail, would not.)
If you have access to the Cambridge Grammar of the English Language (CGEL), this construction is discussed in Chapter 3 ("The verb"), § 4.2.6 ("Future interpretation of a non-deictic present tense"), page 136, as well as in § 4.3.4 ("Non-deictic anteriority"), page 139. The details are surprisingly subtle, but you don't necessarily need to understand all of them to find the explanation and examples useful.
Incidentally, if we "flipped" the sentence from future time to past time, we would use the past tense for all three verbs:
When I completed it, I felt that everything was fine.
This asymmetry between handling of past time and future time is pervasive in English; we arguably don't even really have a future tense, just a past tense, a non-past tense, and various rules for when future time can, must, or must not be indicated in a special way.