First, it is important to be clear on what "most basic form" as described above covers. One notion is "structurally simplest", that is, "having the fewest added things". The other is "phonologically best for predicting other variants". Mixtec seems to qualify as an example of the future being "most basic" because (a) the future has no prefixes or suffixes, (b) certain lexically specified change initial /k/ to [ʃ], and other change tone, in the present tense.
In Logoori, there are 4 future tenses and 5 past tenses; there is no "simple present", there is a present progressive, and a habitual which is a construction build from a hard to gloss particle plus the progressive. The present progressive has a progressive aspectual suffix -a- (a-gʊ́r-áa "he is buying"), and the form does not actually mark the present (it marks the progressive, like English "talk-ing"), since that is how you form the future and past progressives as well.
All verb forms except the imperative have a subject prefix, many have a tense prefix, and all, including the imperative, have a final suffix, either /ɪ/, /i/, /aa/ or, in lieu one of the preceding, /a/. We can exclude the tenses with the special suffixes in the search for the most basic form, since /ɪ/, /i/, /aa/ are non-basic. The candidates are:
Indefinite future: a-ri-gʊ́r-á "he may buy"
Immediate future: a-ra-gʊr-a "he is about to buy"
Most immediate past: y-aa-kʊ-gʊr-a "he just bought"
Recent past: y-aa-ka-gʊr-a "he bought"
Remote past: y-áá-gʊ́r-a "he bought" (long ago)
The imperative wins the competition for having the least morphology. Apart from obvious prefixes, the indefinite future and remote past also have special tone melodies; the imperative does as well, though it is not manifested in the toneless root class that includes /gʊr/.
Since the imperative is usually classed as a mood rather than a tense, and assuming the question is about "which tense", we could strike the imperative from consideration. It then appears that we have one tense that adds two things (the immediate future: a subject prefix and -ra-), and four that add three things (subject prefix and two prefixes, or subject prefix plus one prefix and a tone pattern). The competition seems to favor the immediate future as "most basic" (simplest) – however, this tense, unlike all other forms, cannot be used in a negative clause. With that competitor disqualified, we have a tie between the indefinite future and any of the past tenses, in terms of how many added things they have.
I would favor using the tone-melody as a disqualifier, because tone inflection is really not just one thing, it's an abbreviation for a set of interrelated processes (deletion of the lexical tone, placing another tone somewhere). If we count tone melodies as "more than one", we now have just two forms, the most immediate past and the recent past.