What does the Bible say about cremation?

Actually, the Scriptures do not say anything about required modes of burial for believers.  Standard practice among Old Testament and in the New Testament was burial.  Saul and Jonathan were cremated by the Israelites after their deaths, but this was not normal practice in Israel.  Their bodies were mutilated by the Philistines, thus the decision was made to cremate, then bury the ashes (1 Samuel 31: 8-13).  Achan, and his family, were cremated upon their execution for sinning against Israel, which again appears to be an exception to normal burial practices among the Israelites.

When we realize that all bodies eventually decompose, a body that has been cremated is no worse off!  And of course, we know that God is going to give us new, glorified bodies, whether they have been buried, burned, beheaded, or drowned in the sea.  One positive of burial without cremation is that it does picture Christ's own death, burial, and resurrection which is meaningful in describing our new bodies (Romans 6:3-5; 1 Corinthians 15:3-4).  However, it is not a right/wrong issue. Concern about the cost of funerals is understandable, so be assured that it is within your Christian liberty to choose whichever option you want.