I have done a lot of experimenting on this topic because my body does not fit most road bikes.
A mountain bike cannot become a good roadie bike, but it can become a very fast touring-style bike that will keep up with some (but not all) group rides and let you go on long distance rides with no discomfort. Bikes with no shocks are a bit faster.
By far, the fastest slicks are Tom Ritchey, and I use the 1.4 size so that I don't lose the gearing (smaller diameter would make for a slower top speed).
Advocet are also pretty quick, but any tire that does not skid is slower.
Panaracer Pasela kevlar folding or T-Serv are very fast in sizes 1.25" and smaller. Unfortunately, 1.34" is the minimum size tire for most MTB wheels.
The larger Panaracer sizes are very slow but have extreme traction (any tire that does not skid).
One should not suggest these for speed unless you want all weather speed. These (Pasela, T-serv larger than 1.25") are very fast on rough surfaces (gravel seems not to exist), while a smooth or slippery surface becomes human flypaper. They do stick to wet concrete, even if you really squeeze the brakes--no skid.
While there's no possible better tire for a roadie bike (700c x 28mm T-serv, Pasela kevlar, RuffyTuffy), these probably won't do what you want on a mountain bike.
This is because of the wide width of mountain bike wheels increases traction too much.
If you want speed on a mountain bike, you will have to give up wet weather traction in favor of hard rubber compounds that simply go faster.
It is important to use performance inner tubes (26" road tubes) whenever using 1.4 and smaller tires. Ordinary mountain bike inner tubes will really slow down small performance tires.
Below the 1.4 size, Contenental makes authentic roadie tires in the 1 inch size, but you will need gearing such as 11 tooth on the back and 54 tooth on the front. You will also need rims that are small, such as Sun CR18 or even smaller. Shimano's mega-range freewheels and cassettes (30 to 11) are inexpensive and have the necessary 11 tooth for the back. Nashbar has some in the 8 speed variety. Rocket Ring has the necessary 54 tooth for the front, but there's a catch. If you use the 110 bolt circle gear, there are no pins or ramps to shift quickly. Shimano FC2200 crankset is standard square taper bottom bracket, under $50 and is 130 bolt circle. Sora road double fits the Octalink 2 bottom bracket if you need that. Add $35 for the Rocket ring 130 54 tooth gear and you've almost got it. Almost? Well, a mountain bike front derailer can't shift that combo. You'll need a FC2200, FC2300, Sora, or Tiagra front derailer. Since there's only two front gears in use, an ordinary friction lever will "index" the front two gears just in case your current left-hand shifter won't do it (probably it will work just fine, though).
After going through this, you may discover that you should have purchased a basic road bike instead.
Certainly, do this if you are very short and require a small bike. Otherwise, avoid converting a mountain bike to the 1" tires.
If you just want to go really fast without converting your mountain bike into a road bike, try Schwinn Typhoon tread (26x2.1) tires for extreme low-friction and the big diameter means you don't have to replace your gears. Extreme low friction means hard rubber and that you will wear a helmet in rainy weather. Otherwise, these tires will handle anything (glass, nails, rocks, screws, staples, trees, gravel, fence wire, beer cans). Despite the printing on the side, you can air the front to 40 and the rear to 65. With this size, putting too much air into the front could slow you down, so keep the front 40 or lower. Enjoy the speed, and the really weird looks from you friends (as you pass). This will give you a flat ground pace of 19 MPH. It is extremely enjoyable, but road bikes pace about 21 or so. So, this particular answer gives you all the speed you want for crosstown trips and adventures, but not quite enough speed to join a roadie group ride. *Schwable Big Apple are similar.
For road bike speed, mountain bike handlebars will not work. Honestly, the flat bars or Mary bars do not allow your body to deliver optimal power. However, Wald brand Touring (North Road, smallest 3 speed style, 23 inches) will work just fine. This is an odd-looking way to gain up to 20% more speed. These were first designed to be installed upside down as drop bars, which you can do IF your stem is high enough. If upside down, you can wrap them with roadie bar tape. Also, if upside down, another feature appears. There is both a rearwards grip and a forwards grip that works like drop bars to "duck" you out of the wind without the mandatory view of the front wheel.
If right side up (because your bike is too small or stem is too short, or you just like the classic look), get some very large bmx grips and razor cut the big rubber circles off the ends (makes a big butch look instead of a 3 speed look). Mount the brakes as far forwards as possible without going into the curve. You can still use the forward grip this way (on the Wald version).
The real problem with this style handlebars is getting grips at 1/8 to 1 inch above seat height. Above? Yes. A converted mountain bike becomes a middleweight touring-style bike and will go slow if the handlebars are too low (rider position/power delivery sometimes becomes more important than areodynamics). For Aheadset, there are extender tubes by Delta and severe angle UP stems by Dimension. For 1 inch threadless, Nashbar makes fine adjustable stems. For 25.4mm threaded stems, a dutch-bike adjustable will work (only game in town). For a normal 1" steerer (22.2mm stem), the Pyramid Long Cool stem or Nitto Technomic (25.4mm clamp version) is available in reachs of 80mm for normal people and 100mm for those with long arms or speed freaks (if your bike is the right size, use the 80mm range, but if it is too small for you, get the 100mm range). Lastly, tilt these handlebars so that, when your back is straight, the weight of your body onto your hands is between your thumb and fingers, not on your wrists.
Back on the subject of tires, it simply does not matter what size the tire is, only if the rubber is sticky or hard.
Hard rubber tires are far faster, but a bit outdated because they need to be larger for adequate traction.
Tire weight matters on acceleration, but absolutely does not matter for speed.
In fact, a heavy tire can have a beneficial flywheel effect. Heavy tires can be a problem on group rides because you strain to accelerate, and then ride the brakes as you fly past the pack without pedaling (flywheel effect). Given some speed going down the previous hill (even if you don't pedal), heavy tires will actually coast up the next hill. These are more authentic for touring (commute) and very long distance (Think stomp, stomp, stomp, coast, coast, coast).
Light weight tires will practically fly up a hill, but then they will lose all of your pedaling energy as if by magic and you will also have to pedal down the hill. Still, it is nice to fly up the hill. The same thing happens at a stoplight. With lightweight tires, there's a lot of pedaling and almost NO coasting. This is mostly due to the extra-sticky rubber used in most narrow tires, and it does not apply to all lightweight tires. Tom Ritchey's will coast very well, but they will sometimes skid a bit when you stop. Small tires are more authentic for roadie style and criterion rides (Think spin pedals, spin pedals, spin pedals, because your bike will just stop if you stop pedaling, spin pedals, spin pedals, keep spinning, spin pedals).
Converting a mountain bike into a fast middleweight will let you keep all of the mountain bike features, such as strength, agility, safety and comfort. As such, it is a very worthwhile project.
You can even carry heavy loads with a 19 mph flat ground pace (modern road bikes cannot do that, but the old ones could).
IF your resulting project makes your body comfortable, you will be able to go incredible long distances.
It will not achieve the speed of a basic disposable aluminum road bike such as a Specialized Allez Triple. Those will just toast a mountain bike. So, if your purpose is criterion rides (group roadie rides), purchase a criterion bike.
Even my own middleweight does a great job on distance and blowing the roadies away UP a hill, but after that all I see is a bunch of red led tail lights blinking farther and farther away. So, don't do this project for criterion rides.
However, I really enjoy toasting the mountain bikes and the occaisional extreme speed sprint. That's one other odd feature of a middleweight. They sprint faster than a road bike. Once the sprint is over, well, you know what happens then (you get passed). Out with some friends, I waited patiently until 3 blocks from the house, yelled "See you at the house," slapped the right hand shifter and sprinted like mad. The speedo had just passed 28 MPH before I got tired. That was fun. It wasn't very useful, but it was fun! So, that's what converted mountain bikes and identical middleweights do (sprint and coast). This is an excellent aspect for urban adventure / distance touring, even though it won't work in group rides.
