I have read an article on Wikipedia about irreversible agonists and antagonists. These permanently bind to a target receptor on a cell. However, the receptor protein is then internalized and recycled by the body after the binding occurs. How does the body detect when a protein has been irreversibly bound in such a way? Is there some pathway involved in that or are proteins simply regularly recycled by the body?
The process of downregulating a receptor by internalizing and degrading it in response to (sometimes prolonged) activation or (sometimes prolonged) failure to activate is what pharmacologists call desensitization (in either context). You can read about this generally in Goodman and Gilman's Pharmacological Basis of Therapeutics, Chapter 3, under the subsection "Receptor Desensitization and Regulation of Receptors". There are a number of mechanisms, not all well understood, but the basic idea is that when a signal is constant (either always on or always off), it no longer has any value as a way of providing information. Receptor regulation, or desensitization, is a mechanism for, effectively, resetting the cell or tissue. Removal of the receptor can be followed by recycling, or protein synthesis and return of new receptor to the site.
One well studied mechanism for desensitization is seen in G protein coupled receptors (GPCRs), such as some Dopamine receptors. It is well reviewed here.
To answer your specific question, binding that leads to desensitization is detected the same way any other binding is detected. A receptor produces a signal on binding (or stops producing a signal). The downstream effect of that signal includes a process that involves internalization of the receptor from a location where it can receive signal.
Here that process is mediated by phosphorylation by GRKs (GPCR kinases), which allows for binding of arrestin, and so on...