A great question. We can find the answers from something that happens to us as well as from some experiments that have been done.
First, we know that in some cases, cells from one organ can become established in another organ. Cancer cells do this. Many kinds of cancer that arise in one tissue can grow in other tissues. For instance, liver cancer can cause secondary or metastatic growths in the brain.
In addition there are other examples of tissues moved to solve a condition in another location; mostly things like plastic surgery where normal skin is made into lips, or when veins are used to remake arteries.
But perhaps you are asking about more extremely different types of tissues. Examples include the use of placental membranes to help burn patients grow new skin when they have lost even the ability to make scar tissue.
One of the most successful experiments involving tissue transplanting takes Beta islet cells from the pancreas of normal people and puts them into the liver of type I diabetics. These are the cells that make insulin and they normally grow in the pancreas. The transplanted cells establish themselves nicely in the liver and respond normally to blood sugar levels to make insulin. The cure for diabetes IS right around the corner and is a Canadian invention. The Clinical Islet Transfer Program at the University of Alberta in Edmonton has information on clinical trials. The Edmonton Protocol is described in a news article.
Certainly the success of the Canadian program suggests we may be able to make tissues and organs that can be supported by other tissues or organs to regain lost function. If you are looking at this exciting field for a future career, try bioengineering and tissue engineering as key words in a google search. Both biologists and engineers are required to do these studies and such collaborations will yield exciting finds in the future.