This false teaching takes two verses of the Bible (Mark 16:18 and Luke 10:19) and misapplies them today in an attempt to sensationalize and justify preachers handling snakes. This false teaching often coincides with the “Charismatic” church teachings (e.g., Pentecostals). While it is true the early apostles were gifted with signs and wonders that are no longer active, that doesn’t justify misapplying Scripture to teach or perform snake-handling actions today.
Animals (snakes, serpents, wolves, lions, etc.) are often used in Scripture in a symbolic sense (e.g., Psalm 91:13) as a simile or metaphor; snakes and serpents represent wicked, sinful men and their sinful natures, personality, schemes, and actions. When we “tread” on them, we are overcoming them (their evil plans and actions) by the power of the Word of God through the preaching of the (correct) gospel; we are not literally stepping on the top of a snake. There is no Scriptural basis for touching or handling serpents or snakes today as part of Christianity! 
See also the false teaching on wrongfully interpreting symbolic or figurative language literally.