We can expect different outbreaks. The existing ones might disappear as the host species disappear, as the emerald ash borer wiped out ash species in Toronto. We might have them back in the future.
The gypsy moth is a good example, which Beth brought up. The minimal temperatures over the winter have kept them in check, but as long as we have these higher temperatures over the winter, gypsy moth outbreaks are going to happen more frequently.
Oak wilt is a good example. It has been south of the border for over 40 years. Now it's approaching the border and we have it critically close.
Other pests will probably thrive in these new climatic conditions.