Can "I are" be shortened to "I're"?

I wouldn't use "I're." The usual term is I am, which shortens to "I'm."

But here, the context is "April and I" are...

The "are" doesn't refer to "I" but rather the whole phrase "April and I." That's why it can't be contracted.

You won’t normally find it written, if that’s what you mean. However, you might hear something like it, particularly in rapid speech.

I have Googled for "he and you're", and I found very few examples (most examples have a clause ending with "he", followed by "and you're"). However, "he and you are" is quite common.

Furthermore, Google Ngrams considers "he and you're" so rare it doesn't plot it, while "you're" occurs at two-thirds the frequency of "you are". Similarly, neither "we and they're" nor "they and we're" are frequent enough to be plotted in Ngrams.

Based on these parallel constructions, I would say "you and I're" is very rare and ungrammatical. It appears that, at least in writing, you can't contract the verb when the subject is a compound joined by "and".

Personally, I wouldn't say "he and you're", "they and we're" or "we and they're", either, although I might pronounce "are" as /ər/ in any of these constructions. But notice that /wiər/ (we er) is pronounced differently than /wər/ (we're), at least in my dialect.