A simple, often overlooked cause of this problem is an improperly mixed fuel. Poulan gas chainsaws use a 40-to-1 gas to oil mixture for its fuel. If the ratio of gas inside the mixture is too low, the fuel will be too rich. With an overly rich fuel mix, the engine will ignite the fuel at a lower temperature, and inherently a lower engine speed. Gas that's older than 30 days will also cause running problems, clogging up the fuel that reaches the carburetor. Dump out any old or poorly mixed fuel and resupply it with a fresh batch.
Another simple problem that can cause the engine to shut down after running for a bit occurs when the air filter gets too dirty to draw a sufficient amount of air into the carburetor. The air and fuel needs proper balancing for the fuel to ignite in the cylinder. Pull out the foam element of the air filter and wash it in soapy water, using a mild detergent such as dish soap. Let the filter dry thoroughly before reusing it. A clean air filter will also allow you to properly adjust the carburetor.
The most likely cause of this problem is that the carburetor needs readjusting. The carburetor on a Poulan chainsaw uses three screws to regulate the balance of fuel to air at idle, low and high speeds. Adjust these screws regularly to achieve optimal fuel-to-air ratio and increase engine performance. Even small adjustments to the carburetor can have big impacts on performance. For this reason, Poulan recommends users who don't have experience adjusting the carburetor to take it to a service professional.
If a fresh supply of fuel, clean air filter and adjusted carburetor doesn't solve the problem, you will need to disassemble the carburetor and clean the fuel intake passages. Thoroughly clean the intake needle, lever and bore, and reseat the needle onto the lever. If done poorly, carburetor repairs can damage or ruin the entire Poulan engine. As with carburetor adjustments, if you're not qualified to make these repairs, let a service professional make them for you.