I found exercise to be very helpful coming off the morphine...sweat that shit out....
It is helpful, but for a different reason. A lot of people share the misconception that getting over withdrawal is about sweating it out or excreting drug residues. Really, the great majority of drugs like morphine are excreted relatively quickly. Most of the drug is gone after just 24 hours. Trace amounts of inactive metabolites have no pharmacological effect (don't cross into the brain).
The problem is that the drug has physically caused changes in brain chemistry. In the case of opioids, the brain reduces or stops making endogenous opioid ligands (endorphins, endomorphins, enkephalins) which are the substances that naturally bind to the various opioid receptors in your brain. Other changes including to the receptor sites themselves occur to an extent. Without endogenous opioids, and with your brain re-wired to function on the foreign substance (which you're not giving it), you really feel like shit until your brain learns how to respond and regulate itself on it's own again.
A typical response to exercise is a release of endogenous opioid-like or other reward-enforcing substances in the brain (dopamine).
There is some exciting new research on addiction and gene expression which can hopefully result in more effective treatments at some point in the future.