In which case, it depends on the type of inhibitor. A competitive inhibitor has a structure similar to the substrate, hence would bind to the active site as well, competing with the substrate for the enzyme active sites, decreasing enzymatic activity. A non-competitive inhibitor binds to the allosteric site of the enzyme, causing a structural change in the enzyme active site shape.
There's a difference between the effects of cold-brewed coffee vs. hot-brewed coffee: because different chemical reactions are achieved at higher temperatures, cold-brewed coffee will have a smoother caffeine effect and overall less acidity, which some people experience as part of the kick..
The question is ambiguous in that way, so it could refer to iced coffee or just a cup of hot coffee that sat there for a while and got cold. In the case of a cup of hot coffee cooling off for a while, no, caffeine content will remain the same as it cools.