My Maytag Dishwasher Model #DWC8240ABW will not circulate the water. The spinning arms seem to not spin. Sometimes if I open and shut the door it seems to get them started again. My dishes are not getting clean and the soap is staying in the dispenser.
I have tried using vinegar to clear away any calcium or lime build up. I have taken the spinners off to see if there is any blockage.

This is a very common problem with this unit..
First check the spray arm for any cracks or holes.. Then clean the filter to the pump #6..Then check the impellar assembly #14 make sure the shaft on it is not cracked..(if the impeller is not spining then the shaft may be broken)..
If all checks ok then most likely you have a weak pump..
The pump may be running but not up to speed.. You can also take a glass and turn it right side up in the upper dishrack, run the unit when wash is done if the glass is full of water then you are getting water pressure.
Schematics

Hi MasterTech Thanks for the advice.
Yes the filter was very dirty. I disassembled all the way to the impellar (it is not cracked). I wanted to keep disassembling because I saw another filter that needed cleaning (I believe #16 on the diagram) but was not able to take the impellar off.
Also I noticed on one of the parts #11 that the seal was stretched and did not fit properly. Do you think that is a problem?
My neighbour seemed to think that because the soap dispenser did not open that there might be a problem with the timer/dial, maybe something to do with the wiring?

Yes if your soap drawer is not opening the timer may not be working properly or the linkage in the door is broken..
You would need to remove the door panel to check.. The seal may very well contribute to the problem..
If the screen was clogged cleaning it will help quite a bit..