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.
