The short answer to this is "Yes." Yes, you can install hardwood on top of tile (or natural stone) provided that the tile is intact/good condition (i.e. not coming up/breaking apart), the tile is relatively flat and the hardwood is an engineered hardwood, and given that there is enough clearance in height.
There are 3 ways to install hardwood flooring...and one of them will work on top of tile flooring:
1. Nail down. This requires a plywood sub-floor which is 3/4" thick...or some similar wood surface that is equivalent and has enough structure to secure the nails. The method will obviously not work on top of tile.
2. Glue down. This usually will not work on top of tile either as you need a smooth and somewhat porous surface to the adhesive to adhere to.
3. Float the hardwood floor - this method works on top of tile. This is when the hardwood pieces are attached to each other (usually via a click mechanism) but not directly attached to the floor or sub-floor underneath. Clickable hardwood can be locked into place, similar to how a laminate floor is installed. Underneath the hardwood, you place an foam underlayment so that it is on top of tile, but below the hardwood. This provides a sound barrier and prevents the hardwood from rubbing against the tile (to avoid it wearing down). Hardwood can also be floated if it's glued at the joints. (but usually a clickable hardwood is a bit more secure). A floating floor is secured via the base molding (or shoe molding) around the edges. Both laminate and engineered hardwood floors can be floated.
For more info see: What is a floating floor?
Requirements to install hardwood on top of tile
Please note that floating a hardwood floor on top of tile will not always work. In addition to having a hardwood that can be floated on top of tile, the following factors must also be present: 
1. Tile must be secure. If the tile is not securely adhered to the floor and/or if it is severely crumbling or cracking, installing floor directly on top of it is not a great option. A floor is only as secure as the sub-floor beneath it. If the tile is not secure, it's either best to remove it completely, or pour some sort of self leveling mix or mud on top of it to create a secure and solid sub-floor.
2. The floor should be level and generally even. If the floor isn't even, the floating floor on top of it will move, and it could even crack. It's okay if there are some grout lines where the floor is a bit lower, but if the floor has areas that are higher and lower, the floor will move and shift as you walk on top of it.
3. There must be enough clearance height for doors, cabinets and/or appliances. Check before you install any wood that the doors can open/close. If there isn't enough clearance, then you may need to shave the doors. If they are solid wood doors, this is usually doable, but will cost a bit extra. If the doors are metal, they are extremely challenging to cut, and you may need to get a specialized contractor and/or replace the doors. If they are hollow wood doors, you may need to replace them (because if you cut off the bottom, they may no longer have a bottom. If there is not enough clearance for appliances, you may have some choices to make -e.g. you may need to get new appliances, or leave the existing appliances in place and recognize that when it comes time to replace or fix them, you may also need to replace your floor. It's very important to check these items before you choose your flooring and before you install them. Often, if there won't be enough clearance, it's better to either do a different type of flooring or rip up the tile to address the height issues.
4. It's important that the area is not prone to water, flooding and/or excessive moisture - this would be true for all hardwood floors. When in doubt, it's best to hire a professional. They can advise you on options and recommend the best course. If you live in Westchester County or Stamford CT area and are interested in hardwood floors, feel free to give The Flooring Girl a call at 914-937-2950. (For out of state customers, please call 914-407-3899.)
Related hardwood articles:
- What is a floating floor?
- Can hardwood be installed on top of concrete?
- Can hardwood be installed in a basement?
- What is engineered hardwood?
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