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.
Best Answer
We can't answer that merely from what you have said.
Generally washing machines fall into two types: normal types, which take hot and cold water connections; and heating types, which take cold-only and need more electrical power because they heat water internally. The latter is more expensive but means you don't need to plumb a hot water line.
You can run a normal washer without a hot water line, but it won't clean clothes as well.
Generally dryers fall into two types: Vented dryers, which have a vent to the outside. This is how they remove water vapor from clothing - it's driven out in the air. This also lets them dispose of natural gas exhaust. Or condensing dryers, which remove water using an internal dehumidifier-type device - instead of a vent pipe, they need a water drain connection. Condensing dryers don't have a vent, so they can't be gas.
You can't run a vented dryer without a vent. Seriously, you do not want to do that. You'll be putting gallons of water into the air everytime you run it, which will condense in bad places and rot out your home or spread mold.
Some stacked or partnered washer/dryers also share power supply. For instance one common type uses a 240V/30A connection for the dryer, then the washer gets its electricity from the dryer. This means you don't need a separate power hook-up for the washer.
In some multi-unit developments, the developer selects washer-dryers which are a) condensing, b) heating, and c) shared power supply. This means they don't need to connect a dryer vent, hot water, or 120V power, which saves so much on the building that it justifies the more expensive washer-dryer. If you're in such a building, you need to be aware of this and order compatible washer-dryers.