I have seen an expert commercial pilot pumping the throttle during start for a Cessna 152 and ever since I thought that was ok to do so, but recently studying for ATPL I read about that these technique can produce a risk of engine fire during start. What are the consequences of such a start on short or long term on the engine?
Pumping the carb during a cold start, in an updraft engine such as the 152 has, is the #1 cause of engine fires, and can be eliminated by the proper and factory recommended starting procedure.
The correct procedure is to prime, then open the throtte about 1/4" and then crank the engine. All planes vary, especially if the primers are varnished up to one degree or another, but here is some guidance for a 152 and 172:
10C - 1 full pump of primer
0C - 2 pumps of primer
-10C - 3 pumps of primer
-20C - 4 pumps of primer, however the engine should be preheated prior to starting at this and colder temperatures.
Using the primer, puts a charge in either one or all cylinders, which is adequate to compensate for carb fuel/air mix being condensed on the cylinder walls at the ambient temperature.
Using the throttle pump to enrichen the mixture sprays a little squirt of fuel into the carburetor. If the engine is not running, much of that fuel drips downward out of the "bottom" or inlet side of the carburetor. Too much (easy to do) will drip out, and load the inlet box, and can also drip to the ground. If the engine has even a small backfire, it may ignite the additional fuel there, which will rapidly heat up the carb which has an ample supply of more fuel.
If you can't start with the primer, it is likely that the primers are varnished up and need to be cleaned by a mechanic. They come out, similar to spark plugs.
Also in the 152, you may have a primer on #1 or on all four cylinders. It largely depends on where the plane was initially delivered. If a colder climate, the plane would likely have come with four primers.
Cold addendum #1: It is acceptable on most engines to pump the primer when starting in extreme cold. The way I do this, and normally for -20C and colder starts, is to leave the primer out, and pump it firmly in as the engine starts to fire. Generally no more than three pumps in C172 and C182 aircraft.
Generally, the primer injects into the cylinder, whereas the throttle pump uses the accelerator pump, which injects into an updraft carburetor. It is that large amount of fuel dribbling down the updraft carburetor which creates the starting fire risk.
Again engine configurations vary, and I would talk with a mechanic familiar with cold weather operations to refine your starting procedures. But on aircraft that I have regularly flown, starting at -20C and -30C temperatures is rather routine if your equipment is properly operating, and the crew is properly trained.