Question: A relative asks if he can just put his ancestral tablet (神主牌) in a temple or burn it.
Answer: As ancestral tablets are just focal points for making offerings and prayers, they can be housed in Buddhist temples for doing the same if wished. What most important for the deceased to know is to be mindful of the Buddha (念佛) sincerely with the Three Provisions (三资粮) of Faith, Aspiration and Practice (信愿行), to reach Pure Land swiftly: https://purelanders.com/2018/02/08/the-three-provisions.
There should not be attachment to urns, graves and tablets, or one might become a wandering spirit. If this is already understood personally when alive, having a personal tablet is not needed after death. It is more then, for the living to remember the deceased, and as above, to make offerings and prayers. For this function, even a photo will do, if a focal point is wanted.
Question: He was actually referring to his ancestral tablet at home, with another relative’s name on it.
Answer: How long has the deceased passed away? Are there any supernatural experiences?
Question: Departure was decades ago, and there are no supernatural experiences.
Answer: If rehousing in a temple, it can be moved over respectfully. It is best to recite the following before the tablet, seven times on seven days before moving it: https://purelanders.com/wake. The guidance text (开示文) will have to be tweaked to announce that as the tablet will be moved soon, it should not be attached to. There should only be sincere mindfulness of Buddha, till he is seen, to follow him to his Pure Land (念佛，见佛，跟佛).
If burning, the above should be recited too, before burning it, and scattering the ashes in nature. The guidance text will have to be tweaked to announce that as the tablet will be returned to nature soon, it should not be attached to. There should only be sincere mindfulness of Buddha, till he is seen, to follow him to his Pure Land.
Burning is done as it is difficult for others to reuse the tablet, as it might be assumed to be ‘inhabited’ already, thus ‘inauspicious’. It also reminds all to not be attached to it. Recitation over seven days is to offer ample advanced notice and guidance to reach Pure Land, in case the consciousness of the deceased is still attached or near.
Please Be Mindful Of Your Speech, Namo Amituofo!