Your question isn't limited to FIFOs or any particular interrupt-causing hardware.
Clear interrupt conditions as soon as you know you will handle that condition.
In the case of vectored interrupts, this is very early in the interrupt routine because you know the interrupt cause just from being there. You might do the normal saving of state before anything else, but clearing the interrupt condition should be the next thing you do.
In the case of a shared interrupt routine, you first have to check which device is causing the interrupt. Once you have found a device with its interrupt condition set, you go to the code to handle that device. Clearing the interrupt condition should usually be the first thing this code does.
The main reason for clearing the interrupt condition before servicing the interrupt is to not lose a interrupt. Suppose you did the reverse. The device interrupts, you handle it, then another event occurs right before you clear the interrupt condition. Now the condition for the new event has been cleared, but your code doesn't know that. The state for that device is now hung because it needs service and probably can't generate new events, but the one event that caused the current condition has already been cleared. Oops.
Note that the device you end up servicing in a shared interrupt routine may not be the one that originally caused the interrupt. Multiple devices could have their interrupt conditions set by the time you check them. I generally check them in a fixed order, handle the first one I find that needs service, then get out of the interrupt. If another device also has its interrupt condition set, then the processor will cause another interrupt. Some people go back and loop until no device has its condition set, but this makes the interrupt longer in most cases.
Also be careful checking devices that may have their interrupt masked off at run time. A common example is the UART transmit interrupt. You usually shut that off when the output FIFO is drained, then enable it right after stuffing something into the FIFO. On many processors, masking off the interrupt only prevents the interrupt itself from being generated, not the interrupt condition from being indicated. If some other device causes a shared interrupt and you check the UART transmitter first, you will mistakenly think it needs service since its interrupt condition is set even though its interrupting capability is masked. For any devices that may have its interrupting capability masked off, you have to also check that mask and ignore the device if it is masked off. This may sound obvious, but it is a common rooky mistake.