Yes, you can install a laminate, engineered hardwood, or solid hardwood floor, though solid hardwood expands more and may have trouble with the high heat and/or temperature differentials. Either way, you should acclimatise the flooring by leaving it in the box, in the room where it will be installed for at least a couple days (and a week or two for solid hardwood), before you open the box and install it. If you don't do this, you're likely to have warped boards and/or gaps within a couple days after you install it, effectively ruining the new floor.
You may or may not want to pull up the tiles, as adding another floor on top will add to the height (think of clearance below doors). Height of course depends on what product you choose: laminate can be as thin as 8mm, solid hardwood is usually 3/4".