Can the liver prioritize it's metabolizing potential?
Aside from "toxins", the liver metabolizes a host of endogenous and exogenous compounds. Endogenous compounds are compounds that are normally present and produced within the body like ammonia and estrogens for example. Exogenous compounds are those like ethanol/alcohol, acetaminophen, and other drugs/etc that are not normally in the body but rather we take a mediations or recreational drugs, etc. Most of the compounds that the liver metabolizes are through a set of enzymes called the cytochrome P450 system, which includes numerous enzymes (too many to list here) that catalyze a number of reactions that help detoxify a substance and/or make it more likely to be excreted in the urine by the kidneys.

If the liver "sees" old and new compounds it doesn't differentiate between which it might metabolize first - that is based on concentrations within the liver. In fact, only if the compounds are metabolized by the same liver enzymes will they compete with one another. If they are metabolized by the same exact P450 enzymes, then the concentrations of each of these substances within the hepatocytes (liver cells) will determine which one gets metabolized more frequently. For example, if the concentration of substance A is much, much higher than substance B, then substance A is going to outcompete the other substance for the enzymes that metabolize it (similar to competitive inhibition in biochemistry). Most of the time, endogenous and exogenous compounds do not necessarily compete at the same P450 enzymes because there are so many different P450 enzymes within the liver. 

The liver does not have any way of prioritizing different compounds, however, if the liver is exposed to a particular substance over a long amount of time then it may express more of a particular cytochrome P450 enzyme that metabolizes that particular substance. This is similar to alcoholics being able to metabolize a lot more alcohol than someone who has never drank alcohol previously.