In short it'll shorten the life of the cell. They won't blow up (normally) or fail catastrophically. Most laptop or similar cells can handle up to 5C for short bursts without much effect. But continued use at this rating will destroy them.
Powertool type cells on the other hand, can handle far more current flow. They are normally rated at around 5C continuous, with bursts around 20C or such. Just depends on the chemistry used a particular cell.
It doesn't matter if a single cell delivered 1A, or 20 cells in parallel deliver 20A, the effect is more or less identical to the cells. We could get into differences in internal resistance (IR) and other factors, but we'll leave that for another day