It is very possible for the chemicals to break down or otherwise react with each other to produce chemicals that are less effective or, potentially, more harmful than the original solution.
Many complex chemicals break down into simpler compounds over time and this is often accelerated by exposure to heat or sunlight. Depending on the mixture of chemicals they may well even react to form chemicals that were not originally intended.
This is much more to do with chemical decomposition than computing though.
There may still be enough of the original solution to not affect the performance of the cleaner significantly and I would expect that companies are required to use solutions that are unlikely to have by-products that are harmful to people so chances are you are safe.
But yes, cleaners can "go bad" over time. It may not make it useless though, but we cannot say whether it will now be harmful to plastics or your monitor, though chances are reasonable that it will not.