With or without IT?
 

- They did it in a way that I had never seen (IT?) before.
- We make it move in the way that we want (IT?)to.

Re: With or without IT?
 

first one without 'it' (It's possible to have 'it' in there but it would change the meaning a bit. 'it' inserted sounds like the certain position or arrangement of an object. I think for your question without 'it' would be fine.)

Second one could be both but the meaning is slightly different

- We make it move in the way that we want it to.
(sounds like you are setting up the parameters but 'it' is moving of it's own will or without direct forward motion control by you.)
Here, by inserting it, what you want is 'it to move' in the way you designed.

- We make it move in the way that we want to.
(sounds like you are in complete control of the movement.)

At least that's how it sounds to me. I hope that helps.

Re: With or without IT?
 

Well, for me the first one definitely doesn't have an it. The 'that' puts the emphasis on 'the way', so an it doesn't make sense.

For the second one, I don't think it really matters, but it sounds better English without - no need to repeat.