If you use a portable air conditioner without the exhaust hose then you'll just be dumping the hot air back into your home defeating the purpose of the air conditioner in the first place.
The exhaust hose is needed on the portable air conditioner to exhaust the hot air to the outside so that your home can cool down.
The portable air conditioner cannot work properly without the exhaust hose vented to the outside.
So yes you will need the exhaust hose for the portable air conditioner to work properly and exhaust the hot air outside.