Hi!
No, but can be done with a bypass. So a hardware change is needed to get the Note 3 to boot up without a battery.
The best thing to do is just to buy a new battery and get it in. This will save you a lot of headache and time.
If you do not want to buy a battery, let me know. I can put a website here, explaining how to do it. But I recommend to buy a new battery anyway!
Have a nice day!