They are "irrelevant" in the sense that standard first-order logic does not give us a way to systematically treat them, so in translation these more fine-grained distinctions are usually dropped in order to keep the symbolization simple.
For example, in natural language,
Bill has never eaten a hamburger
we should be able to infer
Bill didn't eat a hamburger yesterday
but this deduction is not possible in first-order logic based on the semantics of the verb tenses alone. We would need special predicates and additional axioms in order to carry out such inferences.
Likewise, in order to express the difference (and non-contradictoryness) between
Bill ate a hamburger
Bill will (not) eat a hamburger
we would have to encode the different verb tenses as two different predicates -- like eat-in-the-past and eat-in-the-future.
Such a symbolization is of course possible, and if you do it this way, there results no contradiction in negating the one of them. But such a more fine-grained symbolization that explicitly encodes tense is usually not done -- and this is what is presumably meant by your source by "verb tense is irrelevant for logic" -- because for the valid arguments that elementary logic can treat, tense does not play a role, since it is not systematically "recognized" by the logic anyway, so we might as well keep the language simple and, in the translation, drop the additional complication of tense that the logic doesn't "understand" anyway.
Whether we say 1. "If Bill is hungry, he eats a hamburger", 2. "Bill is hungry", 3. Therefore "Bill eats a hamburger" or 1. "If Bill is hungry, he will eat a hamburger, 2. "Bill is hungry", 3. Therefore "Bill will eat a hamburger", both inferences can be formalized by symbolizing the different tenses of "eat" as just "eat", and the inference will work in both cases, because the only thing that plays a role for the validity of the argument is modus ponens, and not whether the event expressed by the predication takes place now, in the past or in the future.
The arguments where the difference in time does matter, like the inference presented in my first example, can not be formally treated by standard logic anyway. And for all the inferences that are analyzable with standard logic, it makes no difference whether we talk about now or the past or the future. So for the sake of keeping the language simple and not have overly long predicates we can just translate all of them as "eat" and know as a background assumption that with this translation we mean now or the past or the future.
But that means that when we do want to express more fine-grained distinctions like truth of predications at different points of times, this needs to be done by an explicit (and not very elegant) treatment in the translation by symbolizing the different verb tenses as different predicates that are independent of each other, because FOL itself does not distinguish between different points in time by itself, it only knows predicates and any statement is considered to be evaluated at the same "time".
In order to treat tense and carry out inferences like the one above in a systematic way that more closely reflects how we do it in natural language, we need to enrich the logic and make use of systems like temporal logic or modal logic (or combinations thereof) -- these systems can treat predications under different tenses, but they are not what is usually referred to by ordinary "logic" (which would be standard classical propositional or first-order logic).