Many Protestants believe that once you die you either go to heaven or hell, and our prayers for them wouldn’t change anything at that point. Often prayers at funerals are focused on comforting those left behind, and may mention in a general sense something about God welcoming the departed into His arms. Generally, if the person was a Christian, then any prayers might assume that he/she has gone to Heaven, and usually Bible passages are read that emphasize that… but that’s about it.
When my mom died, the chaplain at the hospital said a pretty general prayer for her just minutes after her death, and thanked God for welcoming her home and releasing her from her suffering. It was a Catholic hospital, but I don’t know if that particular chaplain was Catholic himself.
I think it does come across as insensitive to outright refuse to pray for someone who has died. But I think that’s also why it’s important for us who believe prayers for the dead to be effectual to keep on praying for them – especially for the ones who will have no prayers said for them.
My priest once said that prayer is not limited to time, and that when we pray for the deceased, we don’t know how God will use our prayers. Perhaps our prayers can sort of “reach through time”, and help the departed even during their own lifetimes. For instance, we have an annual requiem Mass at our church for the Cathar martyrs, and even though they’ve been dead for centuries, perhaps our prayers today helped them in their own time, as they descended the mountain to be burned at the stake. Perhaps we are even reaping the benefits of people praying for us in the distant future.