Most single-phase equipment will work fine on three-phase, assuming all other necessary specs are met. You just connect two of the three legs. The load doesn't know there's a third leg and two more phases running around the universe somewhere, any more than your 110V blender knows there's a 220V outlet that your dryer is plugged into. (If you're not in the US and everything is 220V where you are, pretend I didn't say that.) For that matter, your entire house is a single-phase load on a three-phase grid!
Now, the effect the load will have on the source is a different question. Unbalanced loads (loading one phase and not the other two) cause triplen harmonics (3, 9, 15, 21...), which add up as currents in the neutral conductor. This can lead to overheating of that conductor, if it's bad enough. The line distortion can also affect other loads, or cause your bill to go up if the utility cares enough.
As to whether you can run three-phase equipment single-phase, the answer is "maybe, but with derating in most cases". Most three-phase equipment I've seen is either a motor or motor-related. If you try to start a single-phase motor, you'll need external equipment to guarantee that it rotates, and in the direction you want. I've never thought about running a three-phase motor single-phase, but I suspect it may not even rotate. Even if it did, I expect the four windings you'd be using would overheat as well, unless you derated the motor; you can't shove the same amount of power through a third less copper and expect everything to work out the same.
You could use a variable-frequency drive to output three-phase to the motor, and run the VFD off your single phase. But what if the VFD itself was expecting three-phase input? Then if you run it at single-phase, the caps will see 3x the ripple currents they were designed for, and the rectifier will be overloaded (four diodes doing the work of six). So you'd need to oversize your drive, typically by at least 2x, sometimes more. Alternately, you could use an external oversized rectifier with extra caps built in, which is a commercially available product for exactly that circumstance.
Obviously my experience does not cover the entirety of extant three-phase equipment. If you have some other kind of three-phase load in mind, I'd be happy to try to expand this answer to cover it.