Is the DVR supposed to be running continuously?

Yes, this is normal. The HDD is always running and the fan should be on low power. It should not be making very much noise at all however.

Yes, I get the same thing w/my 622. If it gets too annoying, try moving the receiver (or rearrange other components) so the receiver gets a good circulation of airflow. This will cause the fan to not have to run so hard all the time.

So how "green" is this? I would think that something as high tech as a high definition digital video recording satelite TV receiver would be a little more up with the times environmentally. Does anybody know how much power the receiver actually uses when it is "off"?
And what about when we leave town for a while - I would be a little concerned that the thing could overheat and cause real problems.
Is there any "work-around" for this -- for example can I just put a switch between the power souce and the receiver to really turn it off?

Sure, if you don't want to record anything while you are away and you don't want to get Guide updates and you don't want to get software updates, you can power it down. However, be aware that if you leave it off for several days or weeks, you may have a problem when you restart it. It may not get the programming back when it downloads a new Guide. You will have to call Dish and get them to send a signal to the receiver before it will work again.

I had this problem last weekend with an old 311 receiver I don't use much. It had been powered off for 3 weeks and would not work without the signal being sent to it. I thought I had been told in the past it had to be off for at least a month before that would happen. However this time the CSR said it could occur in as little as 4-5 days.

The HDD is always running

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That is definitely a design shortcoming. The HDD on my SA 8300HD DVR spins down a few
minutes after being switched "off". Same with the two Dishplayers I had way back in 2000.