People have been using chimneys for hundreds of years without using Flue liners! You have to determine the condition of the inside of the chimney; maybe the mortar in the brick joints is shot, then maybe a liner would be a good idea.
The chimney breast should never get "very hot", just warmish. Sounds to me it needs a good clean.
Also what are you burning? use seasoned logs or smokeless fuel and NEVER leave the bottom ash-pan door open and forget about it!
Was the soffit plate properly sealed? it could be drawing air through there, hence lack of control of the air flow up the chimney.

Yes, I agree people have been burning open fires for hundreds of years without liners. With an open fires 80% of heat goes straight up the chimney and warm the cold and sometimes damp brick chimney. Wood burners send 25% of the heat up the chimney and 75% to the room. The volume of gas is much less, but exits the burner at more than twice the temperature of an open fire. Sounds like you could have secondary fire in the chimney itself.

I had to get a liner fitted to my own chimney as when I bought the house someone has fitted a stove with just a debris plate with cleaning hatch, you just would smoke myself out of the room trying to get the chimney warmed. The other advantage of a liner is peace of mind. You have far less chance of a leak of dangerous combustion gases into a room above or loft space, less chance of a chimney fire

Did you get a Building Control Certificate from your installer? Without one your're probably no longer insured for fire. Did the installer also fit a sealed CO detector, these became mandatory from October 2010?

With a multi fuel burner the burn a lot hotter than a normal fire ,you must get your chimney swept and a flue installed.new regs require also carbon tester fitted http://www.planningportal.gov.uk/uploads/br/BR_PDF_ADJ_2010.pdf
follow the link and find out more to keep you safe hope this helps David

If its getting that hot I would have a twin wall stainless steel liner fitted, may also consider vermiculite insulation.
Best get a Hetas registered installer to check and advise.
You can now purchase a fan that is made of some form of alloy, it sits on top of stove, no leads etc, when heat transfers through the blades, it causes the fan to spin, sending the heat out, similar to a convector, good bit of kit.

You really need to get a sweep round asap to evaluate the condition of your chimney and i would not use it until this is done

i would certainly have a flu liner put in as your chimney breast should never be too hot.this may be because the joints are weak in the brickwork or it may need a sweep!!
But as plumbing concepts have said it is the peace of mind if you have one fitted.

yes you will need a flue liner if you use the log burner on a regular basis