Is long term continuous use of GPGPU safe for my GPU? I'm looking to use my GPU for non-graphical calculations (artificial life simulations) but ideally I would like to leave this running for weeks at a time, 24 hours a day.

Is there anything I should take into account before trying this? Is a GPU subject to overheating or shortening of its lifespan by continuous use?

What about the computer in general (it's a laptop in my case, but I'm interested in the differences in case it's worth me getting a desktop computer for this)? Will running the GPU non-stop place any strain on connected parts of the computer? Are there known problems that result from this?
Running a GPU at full capacity will reduce its lifespan through electromigration; the speed at which the chips are damaged depends on how hot it is.

A desktop computer has enough room that the designer can put in a cooling system to handle the worst-case thermal situation.  Running your GPU at continuous full load won't overheat it too badly, and you can expect it to last several years.

A laptop, on the other hand, is strongly space-constrained.  Typically, the cooling system is designed to handle brief bursts of heat interspersed with long periods of near-idle operation.  You may not be able to run it at full load 24/7: the BIOS or operating system will slow things down to keep from overheating.  In any case, running at full capacity will likely cause things to burn out in a year or less.