Luan is a relatively cheap plywood made from tropical hardwoods. It is weak, and sometimes bleed oils from the wood. It is not an approved underlayment for ceramic tile.
Multiply plywood may or may not be acceptable. It must be rated Exterior or Exposure 1. The glue laminating the plies must be waterproof. The knotholes in the plies must be plugged. Acceptable grades of plywood are AC, BC or CC (plugged and sanded). To my knowledge, there is no 1/4" thick plywood that meets these specifications.
Installing the plywood over a lineolum floor is OK, but not if it is vinyl. Lino is very hard and not cushioned, but most vinyl sheet flooring is. Cushioned flooring will "give" a little under point loads, this is not a good thing for tile. Are you sure you have honest-to-gawd lineolum?
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