Is using road bike wheels on a mountain bike frame a good idea?

I did this years back with a Giant XTC and some road disk wheels with some 32mm road tyre, very close clearances but it worked surprisingly well ! Toe overlap was a slight issue....but the frame was a little small so figured .....

I'm going to assume the OP has disc brakes, as most rim braked MTB's are an extinct species. 

The main issues then are hub width and keeping the geometry the same with a similar rolling circumference.

So assuming the same, or shimmed hub width......

If 26in wheels: Get some 275 [650b] rims and pop on some narrower slick tyres. The lower profile will take care of the rolling circumference.

If 275/650b: Do like many of the 'Adventure' bikes [Mason Bokeh etc] and mount some 29in/700c rims in the same manner. 

If a 29er: Just swap out the tyres for a narrower road biased tyre - something like a Schwalbe Marathon or Panaracer RibMo [I've had success with the latter] . Getting a narrower road rim and  performance tyre would probably be overkill: In the end, the road performance will be compromised by the MTB geometry. Still can be fun though!

Get some hybrid mtb tyres with a skinny centre and slightly knobbly outer.. such as these http://www.chainreactioncycles.com/schwalbe-furious-fred-evolution-mtb-tyre/rp-prod52617

 I had some Kenda ones a lot cheaper once...job done!

Caveat. I guess I have to say I've never tried fitting 29er wheels to a road frame or vice versa... Maybe that would work, which would make for a pretty quick MTB....

Road bike tyres will not fit on any mtb wheels. Road wheels will not operate on any mtb frame with rim brakes, I don't know  about disc brakes.

However, I have won crits on a (rigid) mtb with 1" slick tyres. A considerable amount of the difference between MTB speed and road bike speed is primarily in the tyres (assuming you can adapt the MTB for a reasonable aero position) 

A set of 1 or 1.25" tyres looks pretty funny on a MTB but they make a huge difference to road riding on a MTB. For speed I go for the narrowest possible. This is a good one. 

http://www.wiggle.co.uk/schwalbe-durano-performance-folding-mtb-tyre-1/

In my experience MTB road tyres last ridiculous amounts of time. I got over 10,000 miles out of one set. 

Could be a good idea - but a few things to think about:

• Do you mean tyres or entire wheels? You mention both. Tyre-only swapping is cheap but a pain and probably saves less time than you'd lose riding knobblies on the road!
• Rim brakes or disc brakes? Discs might make for an easier life since you won't have to worry about the rim being a different width or slightly different diameter (meaning rim brake blocks will the in the wrong place)
• O.L.D. is different between MTB and road frames and rear hubs but you can space out road hubs with a washer either side (put it inside the outermost nuts) or - if you're still to choose your wheels - get ones with MTB width hubs. 

If you are likely to break/wear out your MTB wheels anytime soon, you could just get a new set of MTB wheels and put slicks on them meantime. Or on your existing wheels and get a new set of MTB rims for MTB-ing.

You didn't say if your MTB had 26" or larger wheels though... in the old days, MTB wheels were always 26" (aka 559mm diameter) and wide and road wheels were almost always 700c (aka 622mm) and narrow. Things are more complicated now - road wheels and tyres are often wider and some MTB wheels have larger 622mm diameters, so you might get a good match!

http://forums.mtbr.com/27-5/whats-difference-between-29-tire-700c-tire-6...

A lot would depend on the O.L.D. (Over Locknut Dimension - aka width), of the hubs and your stays. For example most newer road hubs are 130mm while most newer MTB are 135mm.

Sheldon Brown illustration:

I may get excommunicated for this, but if you live somewhere with absolute crap roads, and you're not ready for a cross/gravel bike yet, it's a great idea. It's my current shop bike setup. Get some Schwalbe Marathon City Tyres, or something of the sort. Wouldn't go any smaller than 1.5" (I do 1.75"), or else the geometry is going to start to get funky (also, if you're a bigger guy, then past that the ideal pressure might start to push the rim maximums, FWIW).

PS: Just to clarify, I'm talking about swapping tyres as in the description, NOT wheels as in the topic title. As in, using MTB slicks. 

In answer to your question, no. The length of forks and position of the brake block - rim interface will be at a different place. Skinny road tyres or new mtb wheels with skinny tyres, yes.

Get a spare set of wheels

This is where I bought a lovely pair of wheels from for my MTB, tubeless ready and will take a variety of tyres.

http://www.justridingalong.com/