Is it against the Bible not to have a funeral for a family member who dies? So is a funeral something the Bible requires, either directly or indirectly, if an unbelieving family member dies?
Would it make a difference if the dead person was a Christian, whether to have a funeral or not?
The Bible neither requires nor prohibits the funerals of anyone. In general this is a wisdom issue - would not having a funeral offend the family? Would it damage relationships?
Jesus does however say something that might be taken to say that Christians shouldn't be involved in the funerals/burials of nonchristians:

Another of the disciples said to him, “Lord, let me first go and bury my father.”
And Jesus said to him, “Follow me, and leave the dead to bury their own dead.” (Matthew 8:21-22, ESV)

This shouldn't however be taken as a blanket prohibition of Christians attending funerals for their fathers. These verses, and the ones immediately before them, tell us that there is a high cost to following Jesus, that following Jesus will require us to make sacrifices and that our ties with our families will be strained. And indeed, there are many Christians who have been unable to attend funerals because they are serving God in circumstances that don't permit them to return, such as missionaries in foreign countries.