Is there any positive or negative connotation attached to this phrase?
Suppose I sent someone something by email, and the reply begins:
"I've read your email with interest."
Is that good or bad?
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The sentence means exactly what it says: I read this thing and I found it interesting.
That in itself is generally a positive connotation - things that are interesting are generally better than things that are not interesting.
However the reason for the interest may not be positive. If your boss reads something from you that he expected to agree with his views (which would not have been interesting to him) but in fact it disagrees, that might make it more interesting to him but not necessarily good for you.
However that is entirely from context and can't be deduced from the language.