Can I paraphrase this sentence by "we take the tire from the wheel"?
No, "we take the tire from the wheel" can not be said. "Off" is used here instead of "from" because the tire is on the surface of the wheel, and it is being removed. "From" is generally used when something is being removed that is inside of something. Ex: I took my jacket off when I arrived home. Ex: I removed the cake from the oven.
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