Can human organs be transplanted indefinitely?
First of all, it will be practically impossible to make an estimation of how long a transplanted organ will live and how many times it could be transplanted. However, it is relatively safe to say that it will not live forever. 

To be able to receive an organ transplantation, the HLA MHC-complexes of donor and receiver need to be matched very carefully, to limit the host-vs-graft reaction. This limits the number of possible donor-receiver patient pars drastically. So to be able to do a "row transplantation" (one receiver to the next and so on), they need to have matching MHC-complexes.

Otherwise the immune system of the receiver turns against the transplanted organ and will eventually destroy it. To avoid this reaction, transplantation patients also receive drugs, which severely suppress their immune system, thus making other infections potentially dangerous. This reaction will still take place, since you cannot shut down the immune system of the receiver completely (for obvious reasons), though it happens at a much slower rate. Transplanted organs will only live for a certain time (see estimates from a number of transplantations below) and it is possible that people need to receive transplants twice in their life.

For this reason, no doctor will put their patient at risk by giving them an organ which might have a reduced lifetime. The risk of the operation itself is also substantial.  Additionally, the original receiver needs to be dead to do this, as he would otherise need another organ in exchange.

What we can do is to take the average number of a lot of transplantations together and calculate some average life-years for a transplant. According to table 2 in this paper ("The Life-Years Saved by a Deceased Organ Donor") the "average benefit by type of transplant (life-years)" is:


Kidney: 7.2 years
Liver: 16.9 years
Lung: 2.1 years
Heart: 14.5 years
Kidney-Pancreas: 12.9 years


These numbers are not definite, since the life style of the donor and the receiver as well as infections of the transplated patient (who has a severely suppressed immune system – this is critical) also play an important role in determining the lifetime of the organ.