I have ripped out most of the peas that finished, although I have more than a few odd stragglers which refuse to die and are not showing signs of disease... so I let them be. I plant them for the crop they offer and just as importantly, for their nitrogen fixing properties. Gotta love multiple purpose crops, especially ones that leave the soil in better condition than when they were planted.
Peas are one of those plants that you need to pick frequently to get the most production out of them. They are one of the earlier crops to harvest. Usually the first few rounds of picking for us go right into whatever I am making for dinner. Really sweet ones get quickly blanched and also tossed into salads or ones like sugar snap peas get devoured for snacks.
At a certain point... the family begins to rebel.. and I start blanching and freezing the shelling peas. Snow peas and sugar snap peas get shared with friends and neighbors as I have yet to find a way to put them up that my family will eat later.
Some peas I dry for use in the winter in soups and stews. I pick them when they are wrinkling, but not totally dry and let them finish up indoors. This way they can focus their energy on the immature pods. The ones I leave for seed I do let dry out as much as possible like in the picture above. If I need the space sooner rather than later or the weather does not comply, the whole vine can be cut and hung so the peas can finish drying.
These peas are from the variety Alaska and are fully dried. The smooth dried peas are varieties with a higher starch content than the wrinkled dry peas. Smooth coated seeds are generally the type dried and then used for things like split pea soup.
I try to save at least 2 years worth of plantings in seeds at the minimum, usually I try to aim for enough for 5 years.