I recently bought a used Sears walk-behind mower with a tecumseh engine. It's fairly new, but wasnt used for a while, and wouldnt start for the guy. I bought it assuming I could get it going just by giving it new gas and cleaning out the carb. Not so.
I emptied the gas, gave it new gas, checked the oil, gave it a new spark plug, a new air filter, cleaned the carb, checked the fuel line, checked the spark, all is good. But it still wont start. When I put carb cleaner in the carb itself, or directly into the chamber, it will start for about a second or two then die. One thing I've noticed is that the new spark pulg I put in has oil on it already - it wont start even when its clean, but I wonder is the presence of oil there an indication of what might be wrong?
Any help at all is appreciated - I can't afford to buy a new mower.

Make sure the brass bolt on the carburetor is completely cleaned, all the tiny holes need to be cleaned out or the machine will not run. Oil on the spark plug isn't uncommon for a Tecumseh.

Post engine model number please...

also sounds like it isn't priming, but yes, model number will help greatly

First make sure it is not overfull on oil, it is easy to over fill when the fill tube goes to the top of the engine shroud.
With that said:
Clamp off the fuel line, remove the carb bowl, release the clamp to see if you have good fuel flow to the carb, raise the float to see that it shuts off the flow with minimal pressure, clean the junk from the bowl, clean the bowl nut, 2 holes close to the bottom and a tiny one about midway up(I usually use the wire from a twist tie), spray all holes, cracks and crevases with brake parts cleaner, reassemble and you should be good to go. If this didn't work it will be probably be necessary to rebuild the carb. If I missed something one of the real mechanics will post to help further. Refer to the photo below. Have a good one . Geo
http://i190.photobucket.com/albums/z120/geogrubb/tec_bowl_nut.jpg

Have you done a compression test ? not hard to do,if you are below 15lbs resurface valves,also not hard to do,20 bucks and your good to mow. Remember vac is what hauls the fuel up,and TECH likes to be primed hard 10 to 12 strokes, keep me informed.