The most common non-literal meaning is 'lose interest'. To go limp on something is to lost interest on something. You had a lot of interest or motivation, then suddenly you lose all this interest to continue - you go limp on it.
Always there's the idea you previously had a strong interest on doing something. But that interest suddenly sags, goes limp and you're left with no enthusiasm for continuing.
"I really wanted to go overseas last year, but I went limp on the idea and gave it up"