yes leaving the beans on the plants will stop them from producing more as their primary job to produce seed has been completed.

According to bean expert Gardenlad, you can leave a few for seed and still have the plant continue to make new pods. http://forums.gardenweb.com/discussions/2006996/beans-that-require-trellises?n=7

If what stated in the other thread is true, then would the shell beans have higher production if picked as snap beans?

Also, is it better to save a couple of beans on each plant, or just save a couple of plants for all the seeds? Not sure which one gives more yield.

"If what stated in the other thread is true, then would the shell beans have higher production if picked as snap beans?"

In my experience, if you are referring to beans grown for shellies, probably not. Most of the shell beans I've grown have fewer pods than varieties bred for use as snaps. There are exceptions, such as Goose (Ma Williams) and Brita's Foot Long, which have pretty good yields either way.

For seed saving, my recommendation would depend upon your reason(s) for saving seed... but as a rule, I prefer saving from multiple plants, vice letting a single plant go completely to seed. Beans don't as a rule have a great deal of genetic diversity, but multiple plants helps to preserve more of whatever diversity might be present. It is also good policy to never "put all of your eggs in one basket", so to speak; if that plant becomes diseased or is destroyed by animals, you have lost everything.

If you only want enough seed to replant for your personal use, then just let a few pods ripen on each plant. I've done this with yardlong beans, and they were still going strong when I harvested the dry pods. If you intend to trade, or want larger quantities of seed, you might need to let several plants go to seed. With yardlongs, provided your season is long enough, you can harvest immature pods for several weeks, then let the plants go for seed. This can give you a large seed yield, since the plants will have grown larger during the harvest period, and can now support more pods to maturity. I did this one year with my shortest DTM yardlong (a black-seeded variety) and collected over a pound of dry seed. With yardlongs, because the period from flower to dry is so short, you can have your cake, and eat it too. ;-)

For beans other than yardlongs, the pick-then-let-go method might not be successful, since the period from flower to dry tends to be longer - sometimes much longer. In particular, snap beans bred to stay tender (such as Fortex and Emerite) can take a very long time to mature. If you harvest snaps for several weeks, the beans let go after that might not have time to mature before frost (spoken from experience). So in short-season areas, I would recommend letting the first few pods on each plant go for seed, then harvesting the remainder. Those first few pods are also far less likely to be crossed, since the plants are not yet flowering heavily enough to attract pollinators in large numbers.

Saving seeds gets more complicated.

I have Marc Rogers' book "Saving Seeds". He says to save some, say 10' strong plants for seeds, seeds only. His theory is that, it is better to save the seeds early, not the weak bean pods at the end of season. So we do not take the risk of seeds not mature enough. He did not address the issue with the plants stop producing new beans.

Since I'm almost out of all my bean seeds, this year I want to be at the safe side. So my plan is:

1. Blue Lake, save about 15 plants for seeds only.

2. Dragon Tongue, save 7 plants for seeds only.

3. Chinese Yard Long, save several pods now, pick them when mature, hope they continue to produce.

4. Pebble, Hyacinth and Good Mother Stallard, dry shell bean, keep beans to dry.

Not sure if the above is 100% right, but since I have enough plants, I'd like to experiment with to see if the yield will drop when saving seeds. Still not sure if I can get a good comparison....