A number of folks have reported the same issue. You can find a few threads on the issue here in the discussion groups, each with users who have tried various solutions to no avail:
1) remove all keyboards; hard reset (press home and power buttons for 10 sec); add back emoji keyboard
2) turn off predictive (General -> Keyboard -> Predictive); hard reset; turn on predictive
3) clear keyboard dictionary (General -> Reset -> Reset Keyboard Dictionary); hard reset
none of these have worked.
Some people have mentioned doing a full device wipe; however, I started with iOS 10 on my iPhone 6 out-of-the-box from an Apple Store today, and mine doesn't work. For what it's worth, it doesn't work on my iPad Air 2 either.
The only way I get emoji's is to type the word in question (mac, plane, sheep), then press the keyboard switcher and select the emoji keyboard. This highlights the words in orange from which I can then select emojis. I can't get the predictive selection to work.
I suspect it will end up being a quiet bug fix in iOS 10.0.1 or something like that.