Yes most 3 year old children do need a nap during the day to keep them from becoming too tired and cranky.
My 3 year old would just fall asleep just about anywhere he was when it was nap time and I never had to force the nap most times because he would just fall asleep on his own.
If you notice your 3 year old child is getting upset and cranky it's usually because they need a nap even though they might hate taking a nap.
Sometimes a nap will help keep them from being cranky and I would usually know when it was time for them to nap.
The naps for a 3 year old can happen at anytime during the day but some may need to take a nap for an hour before lunchtime.
If the 3 year old is not tired though then you shouldn't force the issue of taking a nap but if they are showing signs of being tired then a nap would needed.