Are there languages that overtly mark present tense, rather than future/past? In other words, is the present ever more marked?
There doesn't seem to be a way to search for it in WALS, unfortunately.
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(Modern Eastern) Armenian has only periphrastic present tense, for example tesa I saw vs. tesnum em I see. In the present tense there are more morphemes - the ending of the participle and the clitical auxiliary. There is a synthetic future tense, too, but it has an additional morpheme (ktesnem I will see).
Maybe, depending... the main issue is distinguishing marking of aspect vs. temporal reference, and what you mean by "tense". The differences in Classical Arabic between faʕalū and yafʕalūna is sometimes called present / past (tense) and sometimes perfect / imperfect (aspect).
In Anii, there is an unmarked form, and with eventive verbs, it has a past perfective interpretation, not a present or future (they are incompatible with present or future). There is a complication that statives have a present or past imperfective but not habitual reading. There is an imperfective marker tɪ̀ for progressive and habitual and a future marker tɨ́. It may be best to not think of Anii as having specific tense markers and aspect markers, but rather to think of the system as having markers with some tense and mostly aspectual properties. But it does turn out that the unmarked form isn't a present, it's a past.