I've been waiting until I had enough time to give this a proper answer, because the answer to the question in the title is "it depends" and it's a very important "it depends." It runs the gamut from yes to absolutely not. I want to cover road bike tires here as well, as I don't want folks making the jump from "it's ok for mountain bikes" to "it's ok for road bikes." Cyclocross bikes sit right about in the middle depending on the pressure you like to run
For mountain bikes you can run just about any tire you want on a tubeless rim as long as you use sealant. There are some exceptions, as I covered in my answer here. It's also important to note that when you go tubeless you cannot run the high end of your tires' pressure range anymore. The wider the tire, the more this is true. Most folks have no intention of switching to tubeless and pumping their tires up tight, but then again you'd be surprised.
For cyclocross tires, pretty much everything above holds true, especially the bit above pressure. If you're considering running anything above 50psi you need to get a dedicated tubeless cyclocross tire. It's just not worth the risk otherwise.
For road, the answer is an emphatic, unequivocal no. You cannot run road tires made for tubes on a tubeless setup. Ever. They will blow off the rim, hopefully before you even get them up to pressure, and you will get hurt or killed. You can convert a normal rim to a tubeless setup with a kit, but you cannot run a normal road tire on any tubeless road setup. If you want to know why, keep reading...
The whole reason pneumatic tires work is because they're pushing outward on the wall of the tube or tire. PSI stands for pounds per square inch. I assume everyone already knows that but what some haven't thought about is that the term is pretty literal. For every 1 PSI you pump up your tire the air pressure inside is literally exerting a pound of force per square inch on the inside relative to outside atmospheric pressure. That's why if you're a 150lb rider, bike and all, and you have your tires pumped up to 100psi, you've got a grand total of 1.5 square inches of rubber making contact with the road (there are other circumstances to take into account here like casing and compound, but we wont nitpick material physics here). That also means that you've got enormous amounts of pressure trying to tear your tire off the rim in every direction. The bead on wire (non-folding) and kevlar (folding) beaded tires is ever so slightly stretchy. Not very stretchy, but just enough that neither type of bead will stay seated on the rim under the amount of force which the air inside of a pumped up road tire is exerting outward on the tire. Think about what a tube does when you over-inflate it outside the tire- it expands to a much larger diameter than your wheel. Your tire is doing the same thing on a much smaller scale. It's doing this just enough that a standard bead without a tube in place can climb out of the channel where it sits and over the edge of your rim, causing catastrophic results if you're rolling.
With a tube in place inside a rim, the air pressure pushes against the tube first which in turn pushes against the tire which binds the bead in place against the rim. The interaction between the tube pushing against the tire vs just air pressure pushing against the tire is a little different. The tube helps reinforce the pressure against the bead and lock it in place whereas air pressure alone is not enough to fight against the tire's need to expand in diameter.
So how do tubeless road (and "official" tubeless cyclocross) tires fix this problem? Instead of a kevlar or wire bead- both of which have a little bit of stretch to them- tubeless road tires use a carbon fiber bead which has virtually no stretch to it at all. Since they don't stretch, carbon fiber beads are not susceptible to the pressure inside the tire tearing them off the rim like a standard bead (I'll mention that road tubeless tires typically have a little differently shaped bead as well, but this is more to prevent burping when matched with an official road tubeless rim). This is why you can convert a standard road rim to tubeless but you cannot cannot cannot run a standard road tire on a tubeless setup, official tubeless rim or not.
The same problem described for road tubeless can apply to cyclocross or mountain bike setups if the pressure is high enough. That's why carbon fiber beaded tires are available for cyclocross but you can run a standard cross tire tubeless at lower pressure. For mountain bikes non-elastic beads just aren't necessary because no one in their right mind pumps a mtb tire up that tight to begin with- tubeless or not.