Most likely, no. Insects are too well adapted to their environment which means there is little or none selective pressure for them to acquire increases in their intelligence to survive. So unless the environment changes and these changes are such that natural selection can push the arthropoda into developing more intelligence.
Among us (now extinct due to a slight careless with chemistry) vertebrates those species which have developed sapience there is the phenomenon of neoteny where organisms enjoy extended phase of childhood. Adulthood tends to harden the arteries and slow down any capacity for learning and intellectual adaptation. Humans seem to have developed their intelligence because of the massive amount of social interaction we are exposed to and is part of our behavioural heritage.
I agree with TrEs-2b that arthropods like insects and spiders have no capacity of childhood and adolescent learning. They are effectively hard-wired for the behaviour they need for the ecosystems they inhabit and for their survival.
The only tools they have are biological adaptations, eg, web building in spiders and nest building in ants. The only improvement comes from evolutionary changes. No tools in the human sense or tool-making. No tool improvement. No art.
They do communicate via pheronomes and chemical markings. if environmental changes did something to select for enhanced communication this might be the beginning of the evolutionary road to sapience. It's just that this is highly improbable. Very, very improbable. Pity. Smart insects would be a nice idea.
Environmental changes for possible rise of insect sapience
After posting the above answer realisation struck! There will be changes to Earth's environment in the future. Plus the fact that the OP's Evil Overlord has established what are several of the necessary changes for the rise of insect sapience.
In roughly 500 million years time the planet Earth will undergo increased heating as the Sun's evolves along the Main Sequence. This means the future environment will be undergoing massive changes and while insects have been adaptationally stable for many hundreds of millions of years, to survive they will have to undergo radical evolutionary change.
Insects are poikilotherms, their metabolic activity rises with temperature. A hotter planet means more active active insects. This will increase their interaction with their environment thereby increasing selective pressures and driving the increased probability of further adaptation.
However, the first two environmental changes have already been put in place. Competitors with sapience have been removed by the Evil Overlord's poison. Since all large vertebrates are also gone this means any vertebrates with a potential to evolve sapience are also out of the race.
Second, with humans and most of the apex fauna gone plants will thrive and as their abundance rises so will the oxygen content of Earth's atmosphere. A more highly oxygenated atmosphere will enable insects to grow bigger. This means insects will be able to develop bigger brains and comprehensive nervous systems.
If these conditions continue for the next 500 million years as temperatures and environments undergo changes to adapt it is feasible that insects might develop the correct suite of evolutionary adaptations that will enable, at least, insect to acquire sapience.
If so, an insect sapient species may arise under the light of a Sun turned red giant and by the time they develop civilisation this will be under a white dwarf star.
