Can I run a washer hose up and over a door?

No, this will not work. The drain system in a washer is designed to work with a pipe that is about 4' above the floor. This is why they come with a drain hose that will only fit in a drain about that height off the floor.

If you try to make it drain higher up, which is what you are doing, the pump will not be able to push the water that far or, if it is able to, it will be overloaded since it wasn't designed for that purpose.

Let me suggest two options:

1. Move the washer to the area near the pipe.
2. Move the drain pipe to the new location near the washer.

Failing that, the only solution I can think of would be a drain tank next to the washer with a pump that has the capability of draining as you have drawn. But that solution seems like it could be a recipe for disaster over time.

The only exception to this would be if the specification for your particular model of washer says that a lift of 7' is acceptable. But this is going to be the exception not the rule.

EDIT:

You noted in the comments that the manual for the washer says 96". So yes, your plan will work. I'd use PVC pipe but a hose should work as well.

This may work depending on your washing machine. You can test it by holding the hose up high (get a longer one if needed) and see if the machine drains correctly.

In my neighborhood, it's common for washing machines to pump water from basements up to the street level. The reason is our sewer service lines are shallow. Many homes just let that laundry water go out to the storm drains (the city allows it) so their washing machines are pumping the water up around 7 or 8 feet. In my house, we use the sewer for laundry, but the way it's plumbed, the drain is nearly that high. It works fine.