I have broken power supply for PC, 400W. A capacitor has failed. I don't have the same capacitor. The one I want to replace is 220μF 25V, and the one that I have to replace it with is 220μF 16V. Would there be any problem with this?
Capacitor's voltage rating is a very tricky thing. In the event of overvoltage, electrolytics like to fail rather unfortunately. It may simply cause physical electrolyte leakage that renders the cap inoperable. That's why these components are derated in the design stage.
However, since your cap is probably on the 12 Volt rail, there's still a 4-Volt margin. Not much if you ask me (switch mode spikes might be dangerously close to 16V).
Another thing to consider is the ESR rating. Power supplies are very often relying on low-ESR capacitors. Pushing more current through "regular" caps may/will degrade their lifespan.
I'd just go to your nearest electronics supplier and get a 25V lowish-ESR cap.
You absolutely should not replace an electrolytic capacitor with one of lower voltage rating. That would only be okay if the system was designed with a lot of overhead, but for high-volume devices like PC power supplies that's very unlikely. You'll likely end up putting more voltage on your replacement capacitor than it was designed to handle, which could cause a small explosion or a fire.