They can refuse it actually. I learned this just last year the hard way and had to use luggage courrier service or shipping company instead. Even worse, they accepted on the way to but refused when we were going back which left us no choice but to emergency ship which is very costly.
Normally, there is a fee for excess weight up to a certain amount. With Qatar, the fee is per additional kg, depending on the end-points of your journey, and up to a maximum of 32kg per bag. Since there is a fee, they are usually happy to charge you and transport your luggage, except that they may decide that the flight is too loaded and refuse to take it. It happened another time that an airline accepted my excess but sent it on the next flight, so it seems to be up to them.
The best you can do is purchase the excess luggage in advance and that will secure your baggage allowance. Last year, it is when I went to purchase ahead of time that I was not allowed. So if they let you pay for it, you should be pretty much good to go, even if they send it on a later flight. If you cannot purchase the excess in advance, call them as soon as possible and make arrangements if they tell you it will not be possible. They are companies that specialize in shipping luggage but are fairly pricey so also check with generic shipping services. They may have odd requirements. For example, DHL required it to be a box and said if I shipped a suitcase, it had to be boxed while the courrier service only took suitcases and Delta does not access boxes at all airports, so it is possible to go with a box and not be let back on the return journey with it.
