I'm trying to replace the 12V power supply fan in an eMachines EL 1850 desktop computer, due to it becoming too noisy. This is the  original fan, which is rated .27A. I bought  this fan instead, which is rated at .13A, and it seems to maybe have a little weaker blow than the original. Is it safe to use? If the CFM  is not high enough will it be a fire hazard in the power supply, or is there some kind of shutdown mechanism in power supplies that will kick in, if it did overheat? It really depends on the quality of the PSU - MANY PSUs in prebuilt systems aren't good quality and less airflow could well cause problems.  Quality PSUs will shut off if they overheat or if other problems are detected, low quality units don't always have those protection features built in.
Also, the PSU will have a fan curve PROGRAMMED specific to that model - in other words at x temperature or power draw it'll supply y volts to the fan, because that equals z speed.  With a different fan, y volts may equal a totally different speed, or the voltage supplied may even be insufficient to spin the fan up.

In other words, yes it has the potential to cause problems, but depending on how much power you're drawing from the PSU and assuming the fan does work, you MIGHT be OK.  You'll need to test it thoroughly and keep a close eye on things.  It's not something I would advise, a replacement PSU would be a better choice although it would of course cost a little more.Emachines are notorious for cheap flimsy PSU's...
You would be better served REPLACING the WHOLE unit...