how to test electric dryer without 240V outlet?
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.