Yes there can be delays and cancellations, but it depends on the severity of the system faliure.
I was working at EGCC oneday in T3 when there was a power faliure in the new part of terminal, which housed BA. They could do nothing. IIRC there was several cancellations and delays because nothing could be accessed.
When we do manual check in, which takes for ever as you have to hand write the boarding cards and baggage tags and sticker the boarding cards from the manual seat map and then manually board at the gate, it is a nightmare. Manual check in is a last resort if there is no way the systems are going to come back up.
One of the maddest days was in T1 where we had been doing manual check in on all flights. I was checking in a charter and after we closed we headed to the gate to board. We called the flight straight away and no more than 20 passengers turned up. Turned out there had been a bomb scare landside and everybody landside had been evacuated. The flight was very delayed that day.