For me the default is to have the filter engaged. Turning it off is the exception and needs some justification. it usually comes down to two factors: what's being miked and how far it is from the mic.
If the miked instrument contains no useful low frequencies, then the decision is easy. Acoustic guitars, most brass instruments, strings, hand percussion, electric guitars, snares & overheads - none of these things benefit from capturing very low frequencies. All of them are going to HPF'd in the mix anyway.
If it's a human voice, the filter should always be on, IMO. There is never any need to pick up anything below 80 Hz. Those low frequencies are always going to be extraneous noises, pops and proximity-effect woofs. Sure, you can filter them out after the fact, but if you're recording through a compressor you'll definitely want to remove them at the source. They may also determine how much gain you can get from the mic pre, since a low-frequency pop can easily have twice the energy of the main vocal and cause unexpected clipping.
If a microphone had a fixed low-cut filter that I couldn't turn off, I'd be OK with that 95% of the time.