First off, this whole explanation is brought in the Tanya as a way of breaking yourself, intentionally, when you feel like you are distant and cold, when you can't seem to make a connection with HaShem.
So, let's try to envision this in a productive way that might be able to turn this teaching in a way that can inspire us and keep us from getting depressed about the idea of never being able to escape our sins.
In Iggeret haTeshuvah the Baal HaTanya explains that one can think of the six hundred and thirteen mitzwoth as strands of a greater rope that is our attachment to HaShem. Whenever we sin, we damage the related strand, and if our sin is repeated or done with particular disdain, that strand can be broken. No worries though, we still have 612 more strands holding us firmly to HaShem.
Still, because that strand is damaged, our relationship with HaShem is lacking something major. Let's just say it might only be one out of six hundred and twelve, but when each single strand gives us a connection and awareness of the infinite Creator, that's a pretty significant loss.
Also, because this rope links us and HaShem, it transcends time.
Taken this way we can look back on the Baal HaTanya's original comment and understand how it is that our sins are above time and each day, they may cause a lack in our relationship with HaShem anew. Until we retie that strand and re-strengthen it so that it cannot tear again, we run the risk of having to do Teshuvah over and over again. [Presumably this situation described above takes into account Teshuvah from yirah and not from ahavah, which is higher and reseals the breach as if it had never been.]
There remains one more question with our understanding of the Tanya, if it is that David HaMelech discusses this phenomenon as if he has to do teshuvah every day, we cannot assume he never did Teshuvah from Ahavah, but rather I will have to bring the Noam Elimelech to explain how David HaMelech, a Tzaddik, could say that about himself, despite his level.
The Noam Elimelech explains that normal people are bogged down with sin, all the while thinking they are tzaddikim, whereas the Tzaddikim find even their (own) most perfect deeds full of sin and lowliness so that every day the Tzaddik pursues HaShem with renewed intensity and increased vigor. This is a kindness because it inspires the rest of us to recognize our flaws and work to heal them. Here too, David HaMelech, by saying his sin is always before him, inspires the rest of us to seek out our sins and rectify them through ever greater efforts.
The upshot of the Tanya is, on a day where we feel a strong bond with HaShem, we don't need to depress ourselves with these thoughts, and we can safely ignore the Noam Elimelech being beinonim and not Tzadikim ourselves. Still, we can learn from the Tzaddikim that when we feel a strong bond with HaShem and great joy, that we should reinvest that energy into mitzwoth, tefilloth, and Torah study to let HaShem know we appreciate where we are, but we still want more!