A big plot point of Tsukimonogatari revolves around the fact that Araragi has been dipping into his vampire abilities ever since the events of Kizumonogatari that turned him into this half-vampire, half-human... although more human than vampire.
You've seen him using his vampire abilities throughout the series so far, such as when Hitagi staples the inside of his cheek or he fights the monkey spirit possessing Suruga in Bakemonogatari, and when he fights against Kagenui in Nisemonogatari. These are just a few examples.
Every single one of these times, this human was drawing from these inhuman abilities to perform inhuman things. In the time of this story that Tsukimonogatari takes place, it became apparent to Kagenui and Ononoki that this was a problem. If he continued to draw upon these vampire powers like that, it's likely that he could turn into something no longer human.
This is the basis of his "unpaid tabs" line. As Toshinou Kyouko says in their answer, a "tab" refers to a running bill at a bar. At a bar, you can choose to not pay for your drinks, and instead leave a "tab" that you will pay at a later point. Here, the "unpaid tabs" are all the times he used vampire powers. However, at some point, these tabs do need to be paid - namely, with his humanity.
Basically said, it's Araragi (and Nisioisin, the original novel's author) using a metaphor.