At the moment, FIR's greatest weakness is that it is a DIY solution. You will need a PC/Mac/Linux box or a RPi. So you will need some computer skills to get it up and running, particularly if you are running Linux. The other problem is that audio routing is not easy, particularly if you have multiple sources. It works best for a single source, you set it up once, and it works. This is FAR from something like a MiniDSP, where you can switch inputs and adjust the volume with a remote control. And the best thing about a MiniDSP is its robustness, no nightmare software updates or software glitches breaking your DSP chain.
I say "at the moment" because
@mitchco has a
hardware FIR processor in the works. Right now very little is known about it, but if my dreams are fulfilled it will provide MiniDSP-like functionality but with FIR processing.
The other major weakness of FIR is its latency, which is unavoidable. This is dependent on taps/sample rate (NOT on computing power!) and can range between 100ms to 1.5 seconds. This is not really a problem if you are listening to music, but it's a real problem if you need video because of lip sync problems. It can make your system feel sluggish and unresponsive, for e.g. if you change track or adjust the volume, it will happen after a short delay.
As to FIR vs. IIR, FIR is easier to design and more powerful. For e.g. all symmetrical LPF's and HPF's sum to flat, no need to check if certain configurations will have a non-flat sum due to phase interactions. It can do everything IIR can and more. It is the DSP of choice if you like tinkering. And not to mention, it's linear phase. The jury is out whether linear-phase sounds better. I think it does, but that's very much a subjective opinion.