Should it be:
I experience the feeling that you do right now
I experience the feeling that you feel right now
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The first sentence is not acceptable standard English; a feeling is something you might experience, feel, anticipate and so on. Be careful here; feeling in this case is a noun, not a present participle. But a feeling is internal to one person - it is not something you can "do" (which would suggest "doing" to another person).
The second sentence is grammatically correct, but it is usually felt to be poor style, to use such similar words (feeling, feel) together. Instead, a native English speaker would say something like:
I have the same feeling as you.
I am feeling just the same as you are.
I feel just the same as you do.
We can use "do" in that last sentence, because it refers to an action ("feel") of the other person, not a noun ("feeling").