Personally, I do the following:
Tubing: Rinse immediately. Send some hot water through it. Or KMeta solution. Grab in the middle and spin it around to remove most of the water. Then hang dry. Ideally add weights to the ends as the tubing may curl, and the water may not all drain out. Let air dry. Do not leave too long to dry in some times of the year, I have actually had solitary bees try to use the tubing as nests.....
Racking canes, actually plastic is fine. Simply clean them well with KMeta solution and allow to drain. When dry, keep them in a dry area till you need them again.
You can run a string and small bit of cloth through them to clean them even more if you wish.
Both canes and tubes keep out of direct sunlight during storage. Sunlight (mostly UV) can degrade the plastic.
In all cases, re-wash tubes and canes with KMeta solution prior to next use. Inspect the tubes for issues and if none seen, they are most likely fine to use. If you detect issues, simply discard them. Old tubes are great to help support the string around young trees.
Hope this helps.
Side note: I have used the same plastic tubes for over a decade, and almost all are still fine. A few issue here and there, but if following what I wrote above, no worries.