It's a well known limitation but the reason for it is, surprisingly, not common knowledge. Maybe it's a licensing issue?
what's the actual reason that chrome on os x doesn't render emoji?— Isaac Hepworth (@isaach) October 30, 2014
Maybe a strategy tax?
@isaach licensing is the last reason i heard— Juanted House (@capotej) October 30, 2014
Maybe something darker?
@isaach I guess they are pushing their own chrome emoticons?— ☢ Bernardamus ☮ (@Bernardamus) October 30, 2014
@isaach PRISM can't process them yet. Available shortly.— David E (@Eastmad) October 30, 2014
When you think about it, Chrome would need to render Emoji one of two ways: either (a) bundle a set of Emoji with the browser; or (b) use the system set. It's likely true that licensing constraints make (a) difficult. But what about (b)? Apple introduced Emoji to OS X with 10.7 Lion in Summer 2011. Why can't Chrome just render those?
A recent comment on the issue from a Chromium developer notes "The priority over the last year has been DirectWrite and Android, but I'll be getting to this soon".