You can use the tester below. If you touch one probe to the hot and the other to neutral or ground and the bulb should light.
If hot-to-ground lights but not hot-to-neutral the neutral is open. Vice versa for open ground. If you get nothing lighting either both neutral and ground are open or there's no power.
If hot-to-neutral and neutral-to-ground light, hot and neutral are reversed.
If neutral-to-ground and ground-to-hot light, but hot-to-neutral doesn't light, hot and ground are reversed.
http://byramlabs.com/store/images/hi-res/SPE-et-201a.gif
It's a little complicated. Since there's no outlet hooked up you should probably just test to make sure the black-to-white and black-to-green light the light, if not there's a problem further down the line.
I mostly use this kind of tester to see which wires are hot in old work with cloth covered wires that you can't tell which color is which anymore.
It's also a handy tool to have to test all the wires in a box to make sure there's no power in any of them before you start touching them with your hands.
That NEMA 10 dryer receptacle is a hazard all its own, and may be creating a hazard for you right now.
AC power, as implemented in the USA, has many tricky twists that DC people aren't really prepared for. Some are relevant here; I'll describe them.
I gather you know 240V AC power is supplied from a transformer with a center tap. The center tap is defined as "neutral". There is 120V between either phase and neutral. Neutral has one job: be the one and only current return (in DC parlance, negative, Vss, or ground) for 120V circuits.
AC power has a Safety Ground which is tied to earth via water pipe or driven ground rods, and is carried throughout the electrical wiring (except in older wiring). Safety Ground** has one job: to deflect faults away from humans. Ground is not a current return, except during ground fault conditions.
In theory, neutral might "float" above Earth Ground by several hundred or thousand volts, due to capacitive coupling or leakage in the supply transformer. This would be bad, so neutral is "pegged" to Earth/Safety Ground via a tie in the main service panel. This tie is only at the main panel: never in a sub-panel, junction box nor appliance. As a result, neutral is near earth, and any phase is no more than 120V above earth.
The tie has a beneficial side-effect - if a hot wire touches safety ground, it has a fat current path through the grounding system and the tie back to neutral. This completes the circuit, flows high current and trips the circuit breaker. (If a neutral faults to ground, not much happens, as the voltage difference quite small, being only the voltage drop in the wiring.)
Most dryers use 240V for the heating coils, and 120V for everything else. (120V controls are cheaper). So it needs split-phase 240V - both hots and the neutral.
If you've been counting, you know your 3-prong NEMA 10 is a pin shy of a ground. This is where true stupidity sets in. NFPA (the authors of the Electrical Code) compromised with dryer manufacturers -- reasoning that since dryers are rarely ever unplugged or moved, the receptacle isn't likely to fail. So they authorize tying neutral to ground, even though this is dangerous and would be illegal anywhere else. Sigh.
So if you lose/break/float a neutral anywhere between the dryer and the panel, the 120V loads will not have a return, and will "lift" neutral to the voltage of leg A. Voltages will measure 240/0/240 instead of 240/120/120. Since neutral is now "hot" and it's tied to chassis ground, the chassis of the machine is also "hot". This has killed people.
Which is your symptom.
I'd start by replacing that dangerous NEMA 10 receptacle with a modern NEMA 14 connector, which provides a fourth pin for ground. I'd also retrofit a NEMA 14 plug and follow the instruction's (on the web) procedure to separate neutral and ground on the dryer. Also carefully inspect the wires between the receptacle and the panel. Somewhere in here I think you have a faulty neutral.
If you are unable to retrofit a ground wire, then replace the 30A breaker with a GFCI-type breaker, and label the 4-pin receptacle "GFCI Protected/No Equipment Ground" - but if you do that, make triple-darn sure that you remove that dryer's internal neutral-ground bond!
It's also possible your neutral problem is between the panel and the supply, in which case it would affect everything served out of that panel... but I doubt it.
** Electrical Code purists may dislike my careful choice of words "Safety Ground". That is to convey that it isn't anything like Vss. If you really want to know their Legal Names as specified in the NEC, Neutral is "Grounded Conductor" and Ground is "Equipment Grounding Conductor". Say what!? Of course this invites massive confusion, and both are wrong: Neutral is not grounded (except at the main panel), and Ground is not a conductor! (in NEC parlance, only wires which flow current under normal conditions are conductors.) These are terms only their mother could love, and I suggest avoiding them like the plague.
Best Answer
First, a person selling a dryer without providing a 240V connection to demonstrate its performance, will naturally take a significant hit on his resale value.
Any real effort in this regard is going to require the schematic of the dryer and some electrical chops. This forum doesn't have the capacity to convey those chops. Know your skill limits and work within them. With that in mind...
If it has mechanical controls, you could do some "dry" testing by putting an ohmmeter across the various 120V and 240V pins while looking for the expected resistance changes as you operate its controls through their paces. You need to be able scratchpad Ohm's Law a bit to get a sense of what those resistances should be.
If you're willing to open up the controls, you could probe more deeply component by component.
If you want to get "hot", look at the schematic and try to isolate sections that you can test by powering one leg of the 240v, if it makes use of neutral. Be very careful doing that, and positively ground chassis to actual ground, because on many dryers, especially those using NEMA 10/30 connectors, neutral is bonded to chassis -really! - and if you reverse hot/neutral, you can energize the chassis at 120v.
Or by using a 120/240 power adapting transformer - the 40 lb. transformers sold to Euros so they can run their appliances here. You must set the dryer to fluff-only and not operate any heat setting. Transformers are dumb and don't have thermal protection unless it is added, so they would die trying to supply 23A@240V which would be 46A on the 120V side, on wires made for 15A tops. I'm quite sure you'd get a fuse blow or wire fry before you could shut it off.
All this is as dangerous as it sounds and in any case, seems like a lot of work. Depending on how you value your time, it may be "cheaper" to buy enough 10/3 for a temporary run to a workable testing location.