It has to be dried... but as with all roots, I highly recommend you chop it up a bit first... some roots, like echinacea turn to steel when dried. :o)

So if I chop dried roots, then cover them with olive oil,
for how long do I keep them like this? does a jar stand in the sun or dark place ?

dark place... sun will make the oil go rancid, which is always a possibility. You will want as small chopped pieces as you can get, the more to make a 'stronger' oil.

How much dried burdock root and how much olive oil is needed to make an effective hair/scalp treatment oil?
How long does it need to soak?

Yes yo have to use the dried one. You can find lots of info on Green Bare Hands. The cool thing about it is that you can ask them a question and they will get back to you in no time (i needed to know about dry skin and they replied within 1day) their website it www.greenbarehands.com

The best oil is made with the FRESH root. Dig up the root in the autumn after the plant has died down. You want the 1st year root (Burdock is biennial) If you harvest in the spring, you would dig up the second year plant. This is ok but not as potent as the autumn root.

Scrub off the soil but do not use water to wash the root. Chop it into small pieces then let it sit out for 12-25 hours so 'some' of the moisture evaporates. Next fill a dry sterile glass jar with the chopped root and then fill the jar to the top with olive oil. Cover it and let it sit in a cool dark place for 6 weeks. Check on it every few days to release any gas that the infusing root might be producing. Just stir to release these air bubbles.

I'm a herbalist and any solid herbalist will agree that fresh plant oil infusions are far superior than dry. Burdock is so abundant that there is no need to use dry.