Does a research/study have an 'expiry date' that prohibits it from being published in a journal after it has been completed?
Research does not have an "expiry date", but what it does have is competition. Journals want to produce cutting edge research, new information that will help others in the field.
If you do some research but don't publish it, then find 20 years later that no one else has done it and your information is still something other researchers would want to know then you can easily publish it. Delaying publication is only a risk in that other people may work on the same topic or have the same idea you had, and publish it before you.
One interesting example is Brian May, guitarist in the band Queen. He was studying for a PhD when the band took off and quit his studies in 1974 to become internationally famous. In 2007 he re-enrolled at university, completed and submitted his PhD on the work he started 30 years before. Luckily for him, his field was a small one and no one else had decided to work on the topic he had studied in the intervening years, so his work was still new, interesting and relevant.
Most research can be published even after many years. But if the research is out of date by then, the journal may not want it.
An example: I attended a lecture by Chinese mathematician Hua Luogeng sometime around 1980. He mentioned one of his earlier papers, giving the date submitted to the journal and the date published. Then he paused while we all noticed the 10-year-gap between. He explained that the "Cultural Revolution" fits in that gap, when all intellectuals were sent to work on farms in the countryside.
In general, I can not really see how a research will have an expiry date. Maybe if it was "too old" or "proven to be obsolete"? Even then, there might be some merit to your research. This will depend on the way you present you research and market it. After all we publish to "share, help and inform". Some high quality journals may not agree to publish such research, but others with lesser impact factors or standards might.
What is your field? If a research conducted in your field can be labeled as old, perhaps your methods can be applied to other fields. If the results are "very time-dependent" and can no longer be used after a certain date, this can rise some concerns. Also, if your studying an aspect that has been thoroughly studied before, such research might not add much to current literature (not sure if expired is the appropriate term to use for this case).