Is there a statue of limitations as to how long I can continue to request non-payment of rent from a tenant (Oregon)

Not sure what you want to know. If they aren't paying you can get them evicted.
If you mean former tenant. You should have at least a few years, but most renters I know that leave owing don't have the money so even if and when you go through the process and it is ruled in your favor they still might not pay.

If they have moved out, and owe you money, yes there is a statue of limitation on all debts. Not sure the number of years.

If they are still a tenant, why are you not evicting them.

The term Statue of Limitations refers to the time limit one can take legal action against another. So the term has no bearing here. If someone is delinquent in their rent then start eviction proceddings. You can do this as soon as they are 1 day late.