More than half the Tecumseh engined mowers I see that don't start, or won't stay running have gummed up carbs.  Float bowl carbs (B&S too) seem to be a lot more prone to gumming than the B&S Pulsajet (carb on gas tank) carbs.  Here's some things to do when the gas smells or looks like varnish:
Ugh, smells bad, burns your hands, and doesn't run the lawnmower.

So you have to pull the carb off.  Easiest way is to loosen the two screws holding the manifold to the engine.  Then you have to remove the two links to the governor lever without bending them or stretching the spring.  Pay attention to how they're hooked up.  The top lever has a wire that connects to the spring,  The lower one actually moves the carb butterfly.  They cross over so the butterfly link is on the top on the governor lever.  Don't worry, the thirtieth time you do this it will be easy and make sense.  You remember how they were hooked up, right?

This is what a gummed up carb looks like.  Those aren't lichens growing on the float and the inside of the bowl. It needs to come apart (careful!) and be cleaned.  If the carb was leaking as well before it became inoperable, it can mean that either the float has sunk or the needle seat is bad - something that I didn't mess with here.  Do check the float by shaking it (after it's out and clean) - if it rattles it has gasoline inside it and is leaky and needs to be repaired or replaced.  WalMart has great prices on carb cleaner spray, the spray is VERY flammable, and makes a great engine starting fluid too.

This is what the carb body looks like cleaner.  Notice the drilled passage here on the center tube, sometimes they are plugged with gum.

If the carb is gummed up, the carb butterfly probably is stuck too.  That's what the screwdriver blade is pointing at.  Usually a little carb cleaner on the inside and outside will get it moving freely (it has a light spring load) again.

Here's a gummed up main jet.  That's right, it does more than hold the bowl on and there is a small drilling on the unthreaded part of the jet.  There are some slightly different versions, one which has large holes in the side and the main jet is drilled down from the top, but this is the usual type.  This is the main frustration point, because if nobody TELLS you there's supposed to be a hole there, how would you know?  See the little dimple in the picture?

Cleaner.  I have a small copper wire sticking out the main jet!  THAT'S the main reason for it not running because they gum over completely.  Everything I've read says to NEVER use a wire to poke out jets, but I haven't found anything else that will clear a plugged orifice.  Voodoo doesn't work.

 

Some of the newer carbs have EVEN MORE JETS to block up.  You can tell it's one of these carbs by the extra "lump" running up the side of main jet holder in the middle of the carb.  See it?  These carbs have an extra passage that runs from the main jet to the back of the carb by the butterfly.  See the extra jet drill with the wire in it in this picture?  This isn't the main jet, instead the main jet is drilled vertically into the body of the jet and that little area by the larger horizontal drillings trap and HOLD gum and gook to replug the carb. There is also a small drilling in the threads of the jet holder that needs to be cleaned (see the plug about half way up the outside?  The drill is on the INSIDE!)  Then there is a diagonal passage (see that diagonal part of the casting?) that runs to the back of the carb behind the butterfly.  You need to remove the carb from the manifold to get to that.

One other thing to watch out for on these carbs is a rubber o-ring sandwiched up at the top of the main jet holder.  These can go bad and close up.  I found one with a tiny plastic sleeve in the o-ring to keep it open.  When I assembled it with a new o-ring and omitted the plastic sleeve, and found it would not start.

Everything's reasonably clean and ready to go back together.  You checked the float, you hope that the needle seat is good, and you're going to reassemble it JUST like it came apart.  The flat part on the bowl goes toward the float pivot. The float should be just about flat when assembled and you remember how those governor links hooked up, right?

Go wash up, your hands stink and burn from the old gas!