I'm not aware of anything that could really be called an environmental problem in the short term in the Old World. The technology still didn't exist to quickly clear land for new crops, so the issues of the 20th century where woodland clearances led to landslides and local flooding didn't happen in the 1600s.

In the longer term, though, increased food supplies (mostly from the potato) led to increased population, which in turn lead to increased land use, clearance of woodland, reclamation of marginal land, demand for wood and turf for burning, and so forth. In Ireland in the early 1800s, almost every square mile of land was in use, with hillsides terraced to squeeze in more growing space - you can still see the terrace marks in places now, two hundred years later, because nothing's been grown there since except grass for sheep. Some of those led directly into environmental effects.

Vegetable monocultures are a poor foundation for a societal food supply, though, and this became clear in the various famines - again, notably in Ireland. The single potato that likely gave rise to the existence of blight in Europe probably arrived, imported from Mexico via the US, in the late 1830s or early 1840s. You could argue that that was an environmental disaster, but it only affected potatoes and tomatoes, which were of course imports from the New World.

On the other side of things, the introduction of wheat to the New World led pretty directly to the Dust Bowl in the early 20th century, not to mention a swathe of diseases that were introduced to the native populations (not my area of expertise, but I know they were there). And then there was the buffalo hunting. And North European styles of forestry management, which depend on a damper climate in general, lead to forest fires in the drier New World areas.

Sources:

The Columbian Exchange: Biological and Cultural Consequences of 1492, Alfred W. Crosby

The Great Irish Famine, Cormac Ó Grada

S B Goodwin, B A Cohen, and W E Fry, 'Panglobal distribution of a single clonal lineage of the Irish potato famine fungus', Proceedings of the National Academy of Sciences of the United States of America, November 1994, Volume 91, No. 24.