This claim originally comes from East Germany, and specifically from 1950, when a major Potato Beetle infestation was blamed on them being dropped from American planes.

This became a very widespread belief in Eastern Bloc countries (personally I encountered older people saying potato beetles were a CIA plot in Kazakhstan some 15 years ago). It doesn't particularly hold up, though.

First is the issue that while Colorado Potato Beetles are a non-native invasive species in Europe and Asia, they didn't just appear in the 1950s - there had been a recorded presence of them in Germany since the late 19th century, and they were relatively common.

There was an infestation of the beetles in East Germany in 1950, and this threatened the potato crop, which was especially important to East Germans in the aftermath of World War II. But this seems to have been more likely caused by reduced use of pesticides, rather than them being dropped by American planes. The planes idea was somewhat plausible because of the air corridors that were open to American (and British and French) planes flying from West Germany to West Berlin, over East Germany, so there were often American planes overhead (especially during the Berlin Airlift). However, although the British had studied the feasibility of dropping potato beetles on Germany in the First World War, there doesn't seem to be any strong evidence that anyone was seriously considering this during the Cold War.

It did make great propaganda for East Germany, as a lot of ink was split on the campaign against Amikäfer, or "Yank Bugs", and student volunteers were sent out to potato fields to collect/squish them. It absolutely was a constantly-repeated official message that the beetles were there as a result of American sabotage.

This is way less likely for the Soviet Union, by the way. Even if the US was dropping beetles out over East Germany, they weren't regularly flying planes over the USSR, and the ones that were, like the U2s, weren't dropping beetles from 70,000 feet.

Here is a link to a BBC article on the potato beetle battle in East Germany that I am providing because it especially has some great propaganda posters.

Also of interest from a scientific perspective might be this scientific paper "The voyage of an invasive species across continents: genetic diversity of North American and European Colorado potato beetle populations", by Alessandro Grapputo et al. Genetic comparison of North American and European potato beetle populations points to a much lower genetic variability among the European beetles, which would support a history of the European beetles coming from a single founding event (introduction into Europe) in the early 20th century.