The "best for (almost) everybody" approach to this is to NOT check-in for the flight you don't intend to take. This allows the airline to know that you do not intend to fly, and allows them to re-allocate your seat (if needed) sooner - for example, by clearing someone on the standby list. It also means that they won't waste any time waiting for you to turn up at the gate, paging you, etc. If you have checked in they will still be able to re-allocate your seat, but they will normally have to wait until closer to the departure time to do so, when things are already generally fairly hectic!
However there is one potential situation where checking-in might work to your advantage. If the flight you were due to be on is excessively delayed, or if it is canceled, and that delay/cancellation occurs very close to the flight time, then you may stand a better chance of obtaining compensation and/or your money back for the flight if you are checked in.
In some cases even if there is a delay/cancellation you will not be able to do anything simply because you are not at the airport, however many airlines now will allow you to arrange for compensation/refunds over the phone, and sometimes even at a later time. Depending on the exact situation you may be able to do this even if you have not checked in, but depending on the exact situation being checked in will give you slightly more leverage.
There is a second situation where check-in can be beneficial, but it is very much a corner case. If you are going to be at the airport anyway (eg, you are connecting, and the throw-away flight is the connection), AND if the flight you are not planning to fly is oversold, then you may be able to volunteer to be "denied boarding" - ie, give up your seat on that flight and be moved to a later flight, and in doing so receive some form of compensation. The catch with this is that you will need to wait at the gate right up until departure time before they will know that they need your seat so you will need to invest some time, plus if it turns out that they do not need your seat it's going to be rather odd for you to turn around and say you didn't want to fly anyway and not board the plane...
The odds of either of these occurring, and of you being able to benefit from them, is low - but it's not zero. You will need to balance that with the additional inconvenience you will cause the airline staff and potentially other passengers on the flight by checking in.
As for your comment about the airline not being paid until you check in, this is absolutely not true. The airline gets their money regardless of whether you check in or not - in fact most of the time (even when booking via a travel agency) it's the airline themselves that actually charges your credit card, so they have the money from the moment the trip is ticketed.