How long after graduating can I use my former professor as a reference?

There is no hard limit, other than the professor being alive.

Generally, it is useful to include only references that are relevant and can express something about your current abilities. So, if a professor can say something meaningful and relevant about you, even after a decade or more, then you can include them as a reference.

If, however, your abilities and skills have significantly changed in the respective area (e.g. you gained substantially more experience in this particular field), then the reference is outdated, and it is better not to include them.

I have been asked for recommendations from students immediately after a course ends to up to about 5 years later. Some times for graduate school applications, other times for job applications.

To be honest, I see it as part of the job and generally do not mind giving them. I agree that there is not really a hard time limit so long as the recommendation is relevant to your skill set or character. The best thing that you can do is just ask them directly.