Where exactly in your contacts are you trying to paste emojis? I've never had a reason to try it before but I was curious after reading you post so I just tried to do it. I was able to copy and paste an emoji into the name field after my name and then I added a notes field and copy and pasted a different one there. No errors and they saved fine. I have a TMO Note 3 with 4.4.2 using stock keyboard.
Tried it here also (kitkat, verizon) and it works for me too. Stock keyboard as well.
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