Pasta is made of wheat mixed with water, and since wheat is not Paleo, neither is pasta.
The Problems With Pasta
Wheat is one of the biggest offenders to a healthy diet. Not only does it contain gluten, a harmful antinutrient, but it is also extremely high in carbohydrates and low in nutrients. That combination means it causes a spike in blood sugar, followed by a crash. The result leaves you feeling tired, cranky, and craving more carbs to get your blood sugar back up. Once you do, the cycle continues.
Pasta and Insulin
The problem with this cycle is it results in chronically high insulin levels, which leads to insulin resistance. This happens when your cells get so inundated with insulin they stop accepting it. Your pancreas has to produce more insulin to compensate and the cycle continues. This is a dangerous precursor to a number of chronic illnesses, including obesity and diabetes.
Why We Avoid Gluten
Gluten is problematic because it is inflammatory and increases the permeability of the lining of our gut, letting toxins and undigested food pass directly into the bloodstream. The body creates an immune response to deal with these foreign objects, and part of that immune response involves inflammation. Chronic inflammation (just like insulin resistance) is linked to a number of health problems, including everything from high blood pressure to organ failure.
Mark Sisson has a whole list of reasons why grains are unhealthy.