I think the answer to this question will vary greatly by airline.

For example, on AA, check-in time doesn't matter for upgrades (they use request time), but does matter in the case of standby lists and irrops accomodation. The biggie for me is when unexpected traffic or delays get you to the airport later than the checkin cutoff. If you're not checked in, then you're out of luck, but if you've done an online checkin, even w/o printing, then they can still do a boarding pass "reprint". But that's AA: there was a recent long thread on the DL forum indicating that DL would not reprint an never-printed after check-in cutoff time.

As far as downside, on AA, once you check in online, you are no longer able to change seats until you get to the airport. If a better seat opens after you check-in, you're stuck where you are. And if you check in for coach and then your upgrade clears, you can't select your F seat either, until the airport.