Chrome doesn't render emoji. I believe it uses it's own font rendering engine or something along those lines. OSX does render emoji so Firefox shows the character (which I believe relies on the OS font rendering engine).
Granted, this is on Linux, but I think it's only a matter of whether or not your font has glyphs for the emoji characters -- correct me if I'm wrong, but I don't see a technical reason to treat emoji differently from any other Unicode character.
No font can have complete Unicode support (OpenType supports 65k glyphs while Unicode has a little over 100k code points). So font rendering and layout engines usually pick different fonts for different scripts. The reason why Chrome on OS X (and Windows 8) displays Emoji in the title bar but not in the page is likely that the OS knows how to render emoji (in OS X's case even as coloured bitmaps) while Chrome's rendering engine does not. In both cases the emoji are very likely not part of the font (you ship one font that contains the icons and while rendering you can pick them from that font, instead of having to ship every font with those glyphs).
See http://apple.stackexchange.com/questions/41228/why-do-emoji-... for more details.
Also, I am not an expert about this so I could be wrong about the details.