When installing a new tire, it’s a good idea to put the rim on a truing stand to measure radial runout. Check lateral runout at the same to ensure it’s withing 0.80mm. Radial runout is up and down variance from concentric. If the rim has, say 0.80mm or less radial runout, proceed with tire mounting. If there’s more than 0.80mm, a wire spoke rum needs to be re-trued to bring the radial runout within specifications.
A aluminum-alloy cast rim may need replacing depending on how far above 0.80mm the radial runout actually is. Most motorcycle companies advise replacement if radial runout exceeds 0.85mm. It is easy to instruct this, but not everyone has deep pockets with fat wallets, so compromise may same some money.
I do not get worried when replacing a rim for less fortunate customers until hitting 1mm radial runout. I explain the pros and cons and let the customer decide. However, if it goes higher than 1mm, then it becomes a safety issue in my mind, and I will dig my heels and refuse to re-install the rim.