It's a sad state of affairs, this, and there are actually plenty of climate change connections when you start looking...
For example, I'm one of the people monitoring UK Hemiptera species, and we're seeing, on average, four or five new species a year being recorded. We're also getting previous visitors being able to get established, so the total diversity at the moment is actually increasing steadily. There are also changes on a local scale; my area of mid Wales is seeing a lot of new colonists that were previously stuck the other side of the border due to the harsher weather. This has got to be a direct reflection of milder winters, in particular.
The influx of species new to a region has all sorts of knock-on effects, in extreme cases (e.g. the Harlequin Ladybird, or the Elm Bark Borer) having a huge effect on numerous native species. Where a colonist becomes hugely successful, it can knock out some of the ecological buffer systems, leading to boom-and-bust cycles, and unexpected collapses. I'm not sure whether this could result in lower overall insect abundance per se, but it certainly makes the ecology more vulnerable.
The boom-and-bust cycles also result in occasional plagues of aphids, for example, or caterpillars, which are dealt with by farmers in the traditional way: by throwing something toxic at them. This, of course, gets into the local environment, and kills a whole lot more than intended.
A balanced ecosystem, with all its components functioning, is moderately stable and robust to environmental perturbations. An ecosystem with major components suppressed or missing results in more unstable population dynamics, which exacerbates the whole cycle. For example, a small population of caterpillars one year leads to a reduction in their dependent predators; the following year, because the parasitoid wasps have been hit badly and butterflies lay lots of eggs, the caterpillar population booms, and the predators can't keep it under control. So the farmers spray with something extra-nasty, decimating the parasitoids as well as the caterpillars. Rinse and repeat.
I'm sure there are lots of more specific aspects as well. One is changes in weather patterns. In the UK, floods or frost in late April to May are catastrophic for insect populations, because they coincide with flightless larvae; we've had several very bad years. Changes in the plant flowering and budding seasons can also cause problens; last year, for example, I say leaf-feeding insects on a leafless tree, waiting for the buds to burst. Because so many insects are strictly host-specific, a late opening of leaves can result in the population crashing.
There are probably masses more... but insects are extremely sensitive to environmental change. The first clue we had to the rapidity of temperature changes at the end of the last glacial was, I believe, Russell Coope's work on peatland beetles: sub-arctic to Mediterranean faunas in the same place, in a few years. He was soundly laughed at, of course.