What is done with offerings after they've been on the altar awhile?

This answer is indeed in line with Buddhist's precepts. However, this begs the question of food offerings made to the Taoist gods during special occasions like Hungry Ghost and Chinese New Year. I found see shops setting up altars to the Chinese earth gods and offering various type of food, and spotted food thrown out after the offering, hence was wondering if it's spiritually moral to take food offered by others by thanking the gods.

It is also often given as gifts to volunteers or lay members of the temple who can/will use it. It is a gesture of kindness/compassion, as well as giving a blessing to that persons body in the form of food, at least that is how it seems when my Sifu gives fruit to myself and others.

Once an offering is made to the Buddha, it becomes the property of the Buddha, the person who offers it no longer has the ownership. However, Buddha is no longer present to consume them, so the place must be cleaned before things become messy. In Theravada, the common practice is according to the Dakkhinavibhanga sutta commentary. However, I wasn't able to find an English translation of it in internet. So I'm giving the translation as I can. It mentions,

if there's a monk who's an attendant of the Buddha, the offerings should be given to that monk. Other monks are also worthy of receiving them, [as the saying goes] son inherits what belongs to the father. Oil that's left must be used to light the lamps in the temple. Cloths that are left must be used as flags.

If there's no monk in the temple, a lay person who's an attendant can also take them. If there's no such person, there's no harm if anyone else uses them. Nonetheless, if something's offered to the Buddha that can be used in the temple e.g. oil, taking them for personal use can be stealing. Otherwise there's no harm in eating food or even giving them to beggars or animals, you can get more merit :)

That being said, in Dhammadayada sutta, Buddha asks the monks to be the inheriters of his teaching and not the inheriters of his material. Further, he says that the monks who don't eat his leftovers are more praiseworthy than the monks who eat them because it conduces to that monks' few desires, satisfaction, purity, the nature of being supported easily, and for arousing of effort.

...
`Bhikkhus, I have partaken of my meal, is satiated and do not desire any more, there is some morsel food left over to be thrown away, If you desire, partake of it, if you do not partake it I will throw it to some place where there is no green, or I will put it in some water where there are no living things. '

To one of those bhikkhus, it occurs thus: `The Blessed One has partaken of his meal, is satiated and does not desire any more of it. There is some morsel food left over to be thrown away. If we do not partake of it, he will throw it to some place where there is no green, or will put it in some water where there is no life. Yet it is told by the Blessed One this: Bhikkhus, be the heirs of my Teaching and not the heirs of my material. Morsel food is one of the material, What if I spend this day and night bearing up this hunger and discomfort? He not partaking that food spends that night and day bearing up that hunger and discomfort. '

To the other bhikkhu it occurs thus: [repitition] What if I partake of that morsel food and put an end to my hunger and discomfort and spend the night and day?'

So he partakes of that morsel food puts an end to his hunger and discomfort and spends the night and day. Bhikkhus, whoever the bhikkhu who partook of the morsel food, overcame the hunger and discomfort and spent the night and day, yet the earlier bhikkhu is more reverential and praiseworthy. What is the reason? Because it conduces to that bhikkhu's few desires, satisfaction, purity, the nature of being supported easily, and for arousing of effort
MN 3

So I think this is applicable to lay people who are doing the altar maintenance as well.