It depends on the board, etc, and the way your chassis fits onto the motherboard.
Since your motherboard conducts electricity, and so do the screws, its possible if you don't use washers, that the elctricity will flow through the screws, into the chassis, and short it out.
Like you said, you smelt the smell of burning? Hence, probably fried
I'd get some rubber washers and put them inbetween. If it doesnt work then, then I'm afraid you've fried the motherboard and a replacements in order.
Since it was suppsoedly built by proffessionals, they should have took note that it didn't have any, and as a safety procedure, of bought some. I mean, they cost nothing.
At least, alot less than you'll have to fork out now, considering the motherboards probably wasted...
Its worth a try, but that smell couldn't have meant anything good.