Hi, Vicki. You could do it and it would look great--for about a week. The luan is very soft. I think that in a very short time, the surface would start to come up/off in traffic areas. Yes, the varnish would give it some abrasion protection, but since it is so soft, the surface would degrade quickly. The other problem that I see us joining. Luan is only readily available in 1/4" sheets. There is not enough thickness to hold it down and in place. In any case, the joints would be very rough.
Could you do something like this with 3/4" pine or birch plywood? Darned if I know, but you would stand a lot better chance. The thicker pine is also available with tongue and grooved edges. Could work but I would not say for sure what problems you might encounter. I guess that you could also rout T&G grooves on the birch ply.
If you try this, let us know. But I do think that trying it with 1/4" or luan plywood would end up being a big disaster.